序贯高斯模拟在矿石品位估计中的应用研究

随机模拟是地质统计方法的重要内容。在矿石品位估计方法中克里格方法作为一种无偏估计方法,常被用于矿石品位的估计。但克里格法估值存在平滑效应。作者在分析序贯高斯模拟和普通克里格法基本原理的基础上,运用序贯高斯模拟方法和普通克里格方法对某铁矿体内全铁(TFe)品位进行估计,给出了品位估计结果模型。研究从勘探线方向、垂直勘探线方向和竖直方向分别计算变差函数,对球状模型、指数模型、高斯模型的变差函数拟合效果进行了优选,结果表明球型模型拟合效果最好。针对序贯模拟和克里格品位估值效果进行了分析,结果显示:序贯高斯模拟结果在品位分布形态上更接近样品品位分布形态,其平滑效应更小;克里格方法估计与序贯高斯模拟方法相比仅在品位均值方面更接近样本品位均值。因此,认为序贯高斯模拟方法可以更好地刻画矿体内品位分布状态。     

地质统计学中的估值技术和条件模拟

介绍了地质统计学中几种估值技术（线性内插技术、非参数估值技术）和条件模拟技术。在线性内插估值技术中，包括三种基本的克里格估值方法：简单克里金、普通克里金、趋势克里金。非参数估值方法，这里主要介绍指示克里金和广义指示克里金的基本概念和原理。条件模拟技术包括指示函数模拟和退火模拟     

利用地质统计学方法模拟岩石裂隙网络的三维空间分布——以云南个旧高松矿田为例

裂隙在地学的诸多领域中均具有重要意义,其空间分布可以使用地质统计学方法进行模拟,同时考虑裂隙的方向(走向和倾角)。利用序贯高斯模拟方法可以估计裂隙密度的空间分布,并根据裂隙密度数值随机产生裂隙位置的空间分布。裂隙方向被划分成若干(非)均等的方向组,将裂隙方向归属到其所属方向组,表示为由若干二值变量组成的指示形式,0和1分别代表该裂隙方向不属于和属于该组。为了便于计算,减少方向指示变量的成分数目,使用主成分分析法求出方向指示变量的主成分,用普通克里格法估计各主成分的空间分布。把估计结果反演为原始的指示形式,并找出其中数值最大的方向组且将其赋值为1。按照对应方向组内裂隙方向的累积密度函数,随机产生裂隙的方向。根据估计结果,将符合一定距离和角度标准的裂隙元连接为一个裂隙面,从而形成裂隙网络。根据在云南个旧锡矿高松矿田白云岩中进行裂隙网络模拟的应用,可见该方法由于组合了序贯高斯模拟法、普通克里格法和主成分分析法,可以较好地对岩石裂隙位置和方向进行合理的模拟。     

尕斯油田岩性建模算法优选

比较岩性模型建立方法。首先,在高分辨率层序地层学的指导下,最大限度地应用地质、露头、三维地震、测井等静态资料,发挥井点资料垂向分辨率高,地震资料横向信息丰富的优势,在地质规律约束下建立不同时间的高精度等时地层格架模型。然后,在精细格架模型的基础上,以测井解释得到的岩相数据作为条件数据,分别采用指示克里格、截断高斯模拟、Object-modeling算法、贯指示模拟建立砂体展布模型。最后,通过抽稀检验评价不同算法对模拟结果的影响,实现算法及其参数的优选,从而指导整个区块不同开发阶段,不同井网密度时全区三维精细地质模型的建立,也可为具有相似地质环境的油田建立三维地质模型提供参考。通过比较,优选出指示克里格、序贯指示模拟两种算法都能较好表征本研究区地质情况。     

基于不同地质统计方法的渗透系数场对污染物运移的影响

渗透系数场的空间变异性是影响污染物运移结果的决定因素,而地质统计方法是解决渗透系数空间变异性的主要技术手段。本文利用野外场地实测数据,采用普通克里格法和指示克里格法、顺序高斯模拟法和顺序指示模拟法四种地质统计方法,插值估测和模拟再现随机渗透系数场,进而对比研究四种渗透系数场对大尺度污染物运移的影响。研究结果表明,污染羽的质心位置（一阶矩）主要由渗透系数的平均值来决定;污染羽在空间上的展布范围（二阶矩）主要受渗透系数空间变异方差的影响;条件模拟克服了估计法的平滑效果,较好地再现真实曲线的波动性,渗透系数（ lnK）估计方差与污染羽空间二阶矩随着条件模拟次数的增加而减小,并且顺序指示模拟程度更加明显。     

基于二维地质剖面的三维地质结构多点统计学模拟方法

利用三维地质模拟技术重构地质现象的三维空间分布,是实现自然资源管理和风险评估的重要基础和前提。多点统计学方法通过探寻多点间的空间结构关系,结合随机模拟方法生成具有差异性的模拟结果,较好地再现了复杂的地质现象。然而,如何构建合适、有效的训练图像一直是基于多点统计学三维地质模拟的核心问题。本文提出了一种改进的多点统计学算法。本方法结合了序贯模拟和迭代的方法,将二维剖面扩展为三维训练图像,再结合EM-Like算法,实现了三维地质结构的优化模拟。建模实例结果表明,本方法能确保训练图像对内部模拟网格的约束,准确模拟研究区的地层层序,并很好地再现二维地质剖面所反映的地层结构关系。     

干旱区剖面土壤盐分空间变异特征及随机模拟

利用便携式土壤盐分计测定干旱区膜下滴灌棉田及裸地剖面土壤电导率,利用GS+软件确定其半方差函数,并进行序贯高斯模拟来研究土壤剖面盐分的空间变异特征。结果显示:干旱区剖面上土壤盐分空间变异性强,膜下滴灌棉田符合球形模型,裸地区符合高斯模型,且棉田变程远小于裸地;棉田埋深50cm以浅土壤盐分变化剧烈,埋深50cm以深变化平缓,在垂向上呈现3个高值区(30,50和75cm深度附近)及3个低值区(埋深40,60和100cm附近);克里格法具有明显的平滑作用,降低了土壤盐分的空间变异性,而序贯高斯模拟数据更离散,更能反映土壤盐分的空间变异性,但多次模拟均值随次数增加趋于稳定。     

两种方法在地下水位估值中的应用

对于许多区域水资源问题,用数值方法进行潜水水流模拟时,需要给出每个节点上地下水位值。本文首先简单介绍了趋势面方法,然后着重阐述了泛克里格方法的基本原理及它们在地下水位估值中的应用,通过比较两种方法的计算结果可以得出泛克里格方法是进行地下水位估值的空间最优估计方法。     

基于泛克里格方法的位场扩边处理

在频率域进行位场数据处理与转换时,为了减弱边界效应以及Gibbs效应,需要对原始位场数据进行扩边处理,使得网格数据在x、y方向上的点数均为2的幂次方,并且扩边后的边界值均相等。目前比较常用的扩边算法是余弦扩边方法,为了寻求更高的处理精度,这里将泛克里格插值方法引入位场扩边处理中,在频率域计算了位场数据的一阶垂向导数,通过理论模型对比了该方法与传统的余弦扩边以及其他常用网格化方法的频率域处理精度,结果表明:泛克里格方法能很好地实现位场连续且平滑的扩边,并且与其他扩边方法相比有着更高的精度。     

西藏甲玛铜多金属矿床储量的协同克立格估值

西藏甲玛铜多金属矿矿区元素分布复杂,传统的地质统计学方法对其进行储量估算时忽略了多金属的相互影响,因此为了反映出不同金属元素的空间变异情况,采用协同克里格法对该矿区的金属元素储量进行估算。这里首先介绍了协同克里格算法的相关理论和相关技术,然后以此为基础,对不同方向上的空间变差函数进行结构套合的优化,并对协同克里格方程组进行降维处理。最后以2012年甲玛矿区勘探工程的数据为例,以Cu为主区域化变量,以Ag为协同区域化变量,计算了各自的实验变差函数和交差实验变差函数,分别进行协同克里格法插值和普通克里格法插值。交叉验证结果表明,协同克里格估值的标准差为0.6477,在储量计算上面精度更高,并能广泛应用于西藏甲玛铜多金属矿的地质属性、储量估算等空间数据建模。     

克里金算法与多重分形理论在岩土参数随机场分析中的应用

岩土参数的空间分布特征由于存在取样数据之间自相关和互相关的特性,未知点的岩土参数属性可通过特定的方法内插或外推,经典的数理统计方法难以确定周围的数据样本点以及相应的插值系数。首先介绍地统计学中基于距离加权的普通克里金（ordinary kriging, OK）算法、泛克里金算法（UK）和协克里金算法（CK）。由于基于滑动距离加权的OK算法无法度量局部空间的奇异性,将引入多重分形理论弥补该缺陷。以2010上海世博会的世博轴区域（长525 m,宽80 m）为工程背景,区域内共有42个取土钻孔,以典型的粉质黏土层3个重要的物理力学指标,即黏聚力、内摩擦角和压缩模量验证以上算法。对于岩土参数黏聚力和内摩擦角,预测精度由高至低为多重分形联合模型（MK）、协克里金模型（CK）、泛克里金模型（UK）、普通克里金模型（OK）;对于岩土参数压缩模量,相应的顺序为泛克里金模型和普通克里金模型位置互换。研究结果证明,在岩土参数空间场的分析中,辅助信息有助于提高数据预测精度,并且多重分形联合模型有助于分析空间局部的奇异性。     

Object and Pixel-Based Reservoir Modeling of a Braided Fluvial Reservoir

To assess differences between object and pixel-based reservoir modeling techniques, ten realizations of a UK Continental Shelf braided fluvial reservoir were produced using Boolean Simulation (BS) and Sequential Indicator Simulation (SIS). Various sensitivities associated with geological input data as well as with technique-specific modeling parameters were analyzed for both techniques. The resulting realizations from the object-based and pixel-based modeling efforts were assessed by visual inspection and by evaluation of the values and ranges of the single-phase effective permeability tensors, obtained through upscaling. The BS method performed well for the modeling of two types of fluvial channels, yielding well-confined channels, but failed to represent the complex interaction of these with sheetflood and other deposits present in the reservoir. SIS gave less confined channels and had great difficulty in representing the large-scale geometries of one type of channel while maintaining its appropriate proportions. Adding an SIS background to the Boolean channels, as opposed to a Boolean background, resulted in an improved distribution of sheetflood bodies. The permeability results indicated that the SIS method yielded models with much higher horizontal permeability values (20–100%) and lower horizontal anisotropy than the BS versions. By widening the channel distribution and increasing the range of azimuths, however, the BS-produced models gave results approaching the SIS behavior. For this reservoir, we chose to combine the two methods by using object-based channels and a pixel-based heterogeneous background, resulting in moderate permeability and anisotropy levels.     

Estimating Freshwater Acidification Critical Load Exceedance Data for Great Britain Using Space-Varying Relationship Models

In this study, two distinct sets of analyses are conducted on a freshwater acidification critical load dataset, with the objective of assessing the quality of various models in estimating critical load exceedance data. Relationships between contextual catchment and critical load data are known to vary across space; as such, we cater for this in our model choice. Firstly, ordinary kriging (OK), multiple linear regression (MLR), geographically weighted regression (GWR), simple kriging with GWR-derived local means (SKlm-GWR), and kriging with an external drift (KED) are used to predict critical loads (and exceedances). Here, models that cater for space-varying relationships (GWR; SKlm-GWR; KED using local neighbourhoods) make more accurate predictions than those that do not (MLR; KED using a global neighbourhood), as well as in comparison to OK. Secondly, as the chosen predictors are not suited to providing useable estimates of critical load exceedance risk, they are replaced with indicator kriging (IK) models. Here, an IK model that is newly adapted to cater for space-varying relationships performs better than those that are not adapted in this way. However, when site misclassification rates are found using either exceedance predictions or estimates of exceedance risk, rates are intolerably high, reflecting much underlying noise in the data.     

Bootstrap confidence intervals for reservoir model selection techniques

Stochastic spatial simulation allows generation of multiple realizations of spatial variables. Due to the computational time required for evaluating the transfer function, uncertainty quantification of these multiple realizations often requires a selection of a small subset of realization. However, by selecting only a few realizations, one may risk biasing the P10, P50, and P90 estimates as compared to the original multiple realizations. The objective of this study is to develop a methodology to quantify confidence intervals for the estimated P10, P50, and P90 quantiles when only a few models are retained for response evaluation. We use the parametric bootstrap technique, which evaluates the variability of the statistics obtained from uncertainty quantification and constructs confidence intervals. Using this technique, we compare the confidence intervals when using two selection methods: the traditional ranking technique and the distance-based kernel clustering technique (DKM). The DKM has been recently developed and has been shown to be effective in quantifying uncertainty. The methodology is demonstrated using two examples. The first example is a synthetic example, which uses bi-normal variables and serves to demonstrate the technique. The second example is from an oil field in West Africa where the uncertain variable is the cumulative oil production coming from 20 wells. The results show that, for the same number of transfer function evaluations, the DKM method has equal or smaller error and confidence interval compared to ranking.     

Fast wavelet-based stochastic simulation using training images

Spatial uncertainty modelling is a complex and challenging job for orebody modelling in mining, reservoir characterization in petroleum, and contamination modelling in air and water. Stochastic simulation algorithms are popular methods for such modelling. In this paper, discrete wavelet transformation (DWT)-based multiple point simulation algorithm for continuous variable is proposed that handles multi-scale spatial characteristics in datasets and training images. The DWT of a training image provides multi-scale high-frequency wavelet images and one low-frequency scaling image at the coarsest scale. The simulation of the proposed approach is performed on the frequency (wavelet) domain where the scaling image and wavelet images across the scale are simulated jointly. The inverse DWT reconstructs simulated realizations of an attribute of interest in the space domain. An automatic scale-selection algorithm using dominant mode difference is applied for the selection of the optimal scale of wavelet decomposition. The proposed algorithm reduces the computational time required for simulating large domain as compared to spatial domain multi-point simulation algorithm. The algorithm is tested with an exhaustive dataset using conditional and unconditional simulation in two- and three-dimensional fluvial reservoir and mining blasted rock data. The realizations generated by the proposed algorithm perform well and reproduce the statistics of the training image. The study conducted comparing the spatial domain filtersim multiple-point simulation algorithm suggests that the proposed algorithm generates equally good realizations at lower computational cost.     

用协同—指示克立格预测储层物性参数：以胜利油田罗家砂四上砾岩为例

指示克立格是一种非参数统计方法,它能在不舍弃特异值的条件下进行有效的空间估计,协同克立格以协同区域化变量的研究对象,充分考虑了变量空间相关性和变量间的统计相关性,有着单变量克立格法可可比拟的优点,因而,本文提出协同－指示克立格法这一方法,在指示克立格中考虑多变量的空间相关性和变量间的统计相关性,使指示克拉立格与协同克立格相互补充,模拟现实,提高预测精度,并以胜利油田罗家砂四上砾岩为例进行参数预测,     

Conditioning of Multiple-Point Statistics Facies Simulations to Tomographic Images

Geophysical tomography captures the spatial distribution of the underlying geophysical property at a relatively high resolution, but the tomographic images tend to be blurred representations of reality and generally fail to reproduce sharp interfaces. Such models may cause significant bias when taken as a basis for predictive flow and transport modeling and are unsuitable for uncertainty assessment. We present a methodology in which tomograms are used to condition multiple-point statistics (MPS) simulations. A large set of geologically reasonable facies realizations and their corresponding synthetically calculated cross-hole radar tomograms are used as a training image. The training image is scanned with a direct sampling algorithm for patterns in the conditioning tomogram, while accounting for the spatially varying resolution of the tomograms. In a post-processing step, only those conditional simulations that predicted the radar traveltimes within the expected data error levels are accepted. The methodology is demonstrated on a two-facies example featuring channels and an aquifer analog of alluvial sedimentary structures with five facies. For both cases, MPS simulations exhibit the sharp interfaces and the geological patterns found in the training image. Compared to unconditioned MPS simulations, the uncertainty in transport predictions is markedly decreased for simulations conditioned to tomograms. As an improvement to other approaches relying on classical smoothness-constrained geophysical tomography, the proposed method allows for: (1) reproduction of sharp interfaces, (2) incorporation of realistic geological constraints and (3) generation of multiple realizations that enables uncertainty assessment.     

Experimental Assessment of Gradual Deformation Method

Uncertainty in future reservoir performance is usually evaluated from the simulated performance of a small number of reservoir realizations. Unfortunately, most of the practical methods for generating realizations conditional to production data are only approximately correct. It is not known whether or not the recently developed method of Gradual Deformation is an approximate method or if it actually generates realizations that are distributed correctly. In this paper, we evaluate the ability of the Gradual Deformation method to correctly assess the uncertainty in reservoir predictions by comparing the distribution of conditional realizations for a small test problem with the standard distribution from a Markov Chain Monte Carlo (MCMC) method, which is known to be correct, and with distributions from several approximate methods. Although the Gradual Deformation algorithm samples inefficiently for this test problem and is clearly not an exact method, it gives similar uncertainty estimates to those obtained by MCMC method based on a relatively small number of realizations.     

T -distributed Random Fields: A Parametric Model for Heavy-tailedWell-log Data1

Histograms of observations from spatial phenomena are often found to be more heavy-tailed than Gaussian distributions, which makes the Gaussian random field model unsuited. A T-distributed random field model with heavy-tailed marginal probability density functions is defined. The model is a generalization of the familiar Student-T distribution, and it may be given a Bayesian interpretation. The increased variability appears cross-realizations, contrary to in-realizations, since all realizations are Gaussian-like with varying variance between realizations. The T-distributed random field model is analytically tractable and the conditional model is developed, which provides algorithms for conditional simulation and prediction, so-called T-kriging. The model compares favourably with most previously defined random field models. The Gaussian random field model appears as a special, limiting case of the T-distributed random field model. The model is particularly useful whenever multiple, sparsely sampled realizations of the random field are available, and is clearly favourable to the Gaussian model in this case. The properties of the T-distributed random field model is demonstrated on well log observations from the Gullfaks field in the North Sea. The predictions correspond to traditional kriging predictions, while the associated prediction variances are more representative, as they are layer specific and include uncertainty caused by using variance estimates.     

Representing Spatial Uncertainty Using Distances and Kernels

Assessing uncertainty of a spatial phenomenon requires the analysis of a large number of parameters which must be processed by a transfer function. To capture the possibly of a wide range of uncertainty in the transfer function response, a large set of geostatistical model realizations needs to be processed. Stochastic spatial simulation can rapidly provide multiple, equally probable realizations. However, since the transfer function is often computationally demanding, only a small number of models can be evaluated in practice, and are usually selected through a ranking procedure. Traditional ranking techniques for selection of probabilistic ranges of response (P10, P50 and P90) are highly dependent on the static property used. In this paper, we propose to parameterize the spatial uncertainty represented by a large set of geostatistical realizations through a distance function measuring “dissimilarity” between any two geostatistical realizations. The distance function allows a mapping of the space of uncertainty. The distance can be tailored to the particular problem. The multi-dimensional space of uncertainty can be modeled using kernel techniques, such as kernel principal component analysis (KPCA) or kernel clustering. These tools allow for the selection of a subset of representative realizations containing similar properties to the larger set. Without losing accuracy, decisions and strategies can then be performed applying a transfer function on the subset without the need to exhaustively evaluate each realization. This method is applied to a synthetic oil reservoir, where spatial uncertainty of channel facies is modeled through multiple realizations generated using a multi-point geostatistical algorithm and several training images.