Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics

In many earth sciences applications, the geological objects or structures to be reproduced are curvilinear, e.g., sand channels in a clastic reservoir. Their modeling requires multiple-point statistics involving jointly three or more points at a time, much beyond the traditional two-point variogram statistics. Actual data from the field being modeled, particularly if it is subsurface, are rarely enough to allow inference of such multiple-point statistics. The approach proposed in this paper consists of borrowing the required multiple-point statistics from training images depicting the expected patterns of geological heterogeneities. Several training images can be used, reflecting different scales of variability and styles of heterogeneities. The multiple-point statistics inferred from these training image(s) are exported to the geostatistical numerical model where they are anchored to the actual data, both hard and soft, in a sequential simulation mode. The algorithm and code developed are tested for the simulation of a fluvial hydrocarbon reservoir with meandering channels. The methodology proposed appears to be simple (multiple-point statistics are scanned directly from training images), general (any type of random geometry can be considered), and fast enough to handle large 3D simulation grids.     

基于随机游走过程的多点地质统计学建模方法

在分析国内外建模方法现状及其特点的基础上,提出了一种用于河流相储层模拟的新方法,即基于随机游走过程的多点地质统计学方法(RMPS).首先提出了7个方向迁移概率计算及4个方向河道源头搜索的随机游走过程的改进,实现了高曲率回旋河道和网状河等模拟以及各种类型河流相的主流线预测.其次在预测河道主流线的基础上,利用它对多点地质统...     

Filter-Based Classification of Training Image Patterns for Spatial Simulation

Multiple-point simulation, as opposed to simulation one point at a time, operates at the pattern level using a priori structural information. To reduce the dimensionality of the space of patterns we propose a multi-point filtersim algorithm that classifies structural patterns using selected filter statistics. The pattern filter statistics are specific linear combinations of pattern pixel values that represent directional mean, gradient, and curvature properties. Simulation proceeds by sampling from pattern classes selected by conditioning data.     

The Necessity of a Multiple-Point Prior Model

Any interpolation, any hand contouring or digital drawing of a map or a numerical model necessarily calls for a prior model of the multiple-point statistics that link together the data to the unsampled nodes, then these unsampled nodes together. That prior model can be implicit, poorly defined as in hand contouring; it can be explicit through an algorithm as in digital mapping. The multiple-point statistics involved go well beyond single-point histogram and two-point covariance models; the challenge is to define algorithms that can control more of such statistics, particularly those that impact most the utilization of the resulting maps beyond their visual appearance. The newly introduced multiple-point simulation (mps) algorithms borrow the high order statistics from a visually and statistically explicit model, a training image. It is shown that mps can simulate realizations with high entropy character as well as traditional Gaussian-based algorithms, while offering the flexibility of considering alternative training images with various levels of low entropy (organized) structures. The impact on flow performance (spatial connectivity) of choosing a wrong training image among many sharing the same histogram and variogram is demonstrated.     

Conditional Simulation with Patterns

An entirely new approach to stochastic simulation is proposed through the direct simulation of patterns. Unlike pixel-based (single grid cells) or object-based stochastic simulation, pattern-based simulation simulates by pasting patterns directly onto the simulation grid. A pattern is a multi-pixel configuration identifying a meaningful entity (a puzzle piece) of the underlying spatial continuity. The methodology relies on the use of a training image from which the pattern set (database) is extracted. The use of training images is not new. The concept of a training image is extensively used in simulating Markov random fields or for sequentially simulating structures using multiple-point statistics. Both these approaches rely on extracting statistics from the training image, then reproducing these statistics in multiple stochastic realizations, at the same time conditioning to any available data. The proposed approach does not rely, explicitly, on either a statistical or probabilistic methodology. Instead, a sequential simulation method is proposed that borrows heavily from the pattern recognition literature and simulates by pasting at each visited location along a random path a pattern that is compatible with the available local data and any previously simulated patterns. This paper discusses the various implementation details to accomplish this idea. Several 2D illustrative as well as realistic and complex 3D examples are presented to showcase the versatility of the proposed algorithm.     

中国北方岩溶区含水岩溶裂隙介质的序列指示模拟研究

针对当前岩溶学研究领域中空隙位置定量预测和数学描述的难题，选择研究程度高、资料丰富、具有典型代表性的山东济南泉域的含水岩裂隙介质，运用序列指示条件模拟方法研究空隙间的连通性能，预测岩溶裂隙密集带和强径流带的位置。研究表明，它既可给出其模拟数值的波动区间，朋不同分位数值模拟图均能明显一致地指示出岩溶裂隙密集带和强径流带位置，尤其是当样本点较少时，也可获得几乎与样本点较多时的一致成果，从而为该方法的推广应用提供了广阔的领域。     

Implementation schemes for avoiding artifact discontinuities in simulated annealing

The simulated annealing algorithm has been applied successfully to conditional simulation of categorical variables (e.g., rock or facies units) with the objective of improving the match between measured and modeled spatial variability. In some implementation schemes, however, spurious features termed “artifact discontinuities” may occur near conditioning data, especially during the “zero- temperature” case referred to as simulated quenching. This paper shows that artifact discontinuities can be avoided by considering the anisotropy of the spatial variability model, reducing the number of lag vectors used in the objective function, and providing a rudimentary initial configuration. Results from several test cases suggest that the artifact discontinuities might be caused by overly precise fitting of measured to modeled spatial variability.     

Multiple-Point Simulations Constrained by Continuous Auxiliary Data

An important issue of using the multiple-point (MP) statistical approach for reservoir modeling concerns the integration of auxiliary constraints derived, for instance, from seismic information. There exist two methods in the literature for these non-stationary MP simulations. One is based on an analytical approximation (the “τ-model”) of the conditional probabilities that involve auxiliary data. The degree of approximation with this method depends on the parameter τ, whose inference is difficult in practice. The other method is based on the inference of these conditional probabilities directly from training images. This method classifies the auxiliary data into a few classes. This classification is in general arbitrary and therefore inconvenient in practice, especially in the case of continuous auxiliary constraints. In this paper, we propose an alternative method for performing non-stationary MP simulations. This method accounts for the data support in the modeling procedure and allows, in particular, continuous auxiliary data to be integrated into MP simulations. This method avoids the major limitations of the previous methods, namely the use of an approximate analytical model and the reduction of the auxiliary data into a limited number of classes. This method can be easily implemented in the existing MP simulation codes. Numerical tests show good performance of this method both in reproducing the geometrical features of the training image and in honouring the auxiliary data.     

Improving Sequential Simulation with a Structured Path Guided by Information Content

Multiple-point simulation is a newly developed geostatistical method that aims at combining the strengths of two mainstream geostatistical methods: object-based and pixel-based methods. It maintains the flexibility of pixel-based algorithms in data conditioning, while enhancing its capability of reproducing realistic geological shapes, which is traditionally reserved to object-based algorithms. However, the current snesim program for multiple-point simulation has difficulty in reproducing large-scale structures, which have a significant impact on the flow response. To address this problem, we propose to simulate along a structured path based on an information content measure. This structured path accounts for not only the information from the data, but also some prior structural information provided by geological knowledge. Various case studies show a better reproduction of large-scale structures. This concept of simulating along a structured path guided by information content can be applied to any sequential simulation algorithms, including traditional variogram-based two-point geostatistical algorithms.     

多点地质统计学建模中训练图像建立方法综述

多点地质统计学建模是近年来储层建模技术的研究热点,其实用性受到训练图像的限制。训练图像的质量决定了多点地质统计学建模的精度和可靠程度,是多点地质统计学建模成功的关键因素。文章阐述了训练图像的特征和意义,从方法的定义、使用情况、实例等方面系统介绍了训练图像建立的方法,包括手工绘制、基于目标模拟、三维地震信息提取或转化、基于原型模型、基于过程模拟和二维图像方法,综合对比了不同训练图像建立方法的数据来源、优势与不足,探讨了多点地质统计学建模依赖训练图像存在的问题。结合文献调研和多点地质统计学建模实践,指出了训练图像及其建立方法的发展方向,为多点地质统计学建模研究者和使用者提供借鉴,为完善多点地质统计学建模方法提供思考。     

Evaluating Thickness of Bauxite Deposit Using Indicator Geostatistics and Fuzzy Estimation

Evaluating the geological properties of a mineral deposit is a fundamental task for mine planning and it requires an assessment of reserve parameters such as thickness and grade. This paper presents a linguistic model for estimating bauxite thickness within a deposit in a fuzzy environment using indicator geostatistics and fuzzy modeling. The proposed model consists of two main stages: determining the orebody boundary and estimating the thickness. In order to estimate the thickness, a rule‐based fuzzy inference mechanism has been developed based on data statistics. Results and performance of the model have been compared with that of a well‐known conventional technique, geostatistics (kriging), and it is shown that the proposed model has bigger estimation power. In addition, the fuzzy approach is more flexible than the kriging approach. The fuzzy methodology used in the present paper is convenient for modeling reserve parameters.     

Conditional Simulation of Nongaussian Random Functions

This paper presents a conditional simulation procedure that overcomes the limits of gaussian models and enables one to simulate regionalized variables with highly asymmetrical histograms or with partial or total connectivity of extreme values. The philosophy of the method is similar to that of sequential indicator technique, but it is more accurate because it is based on a complete bivariate model by means of an isofactorial law. The resulting simulations, which can be continuous or categorical, not only honor measured values at data points, but also reproduce the mono and bivariate laws of the random function associated to the regionalized variable, that is, every one or two-point statistic: histogram, variogram, indicator variograms. The sequential isofactorial method can also be adapted to conditional simulation of block values, without resorting to point–support simulations.     

A neural network approach to geostatistical simulation

Neural networks offer a non-algorithmic approach to geostatistical simulation with the possibility of automatic recognition of correlation structure. The paper gives a brief overview of neural networks and describes a feedforward, back-propagation network for geostatistical simulation. The operation of the network is illustrated with two simple one-dimensional examples which can be followed through with hand calculations to give an insight into the operation of the network. The convergence of the network is described in terms of the variogram calculated from the values at each of the output nodes at each iteration.     

An Improved Perturbation Mechanism for Simulated Annealing Simulation

Simulated annealing (SA) is being increasingly used for the generation of stochastic models of spatial phenomena because of its flexibility to integrate data of diverse types and scales. The major shortcoming of SA is the extensive CPU requirements. We present a perturbation mechanism that significantly improves the CPU speed. Two conventional perturbation mechanisms are to (1) randomly select two locations and swap their attribute values, or (2) visit a randomly selected location and draw a new value from the global histogram. The proposed perturbation mechanism is a modification of option 2: each candidate value is drawn from a local conditional distribution built with a template of kriging weights rather than from the global distribution. This results in accepting more perturbations and in perturbations that improve the variogram reproduction for short scale lags. We document the new method, the increased convergence speed, and the improved variogram reproduction. Implementation details of the method such as the size of the local neighborhood are considered.     

阿舍勒铜矿可视化储量计算的指示克里格法应用研究

应用计算机图形学方法在VC.net环境下利用Open GL图形库函数完全自主开发了指示克里格法的三维可视化储量计算模块,给出了指示克里格法的计算步骤,并设计出阿舍勒铜矿体的指示克里格法可视化储量计算流程,实现了对阿舍勒铜矿体的指示克里格可视化储量计算。结果表明,指示克里格法储量计算能够取得较理想的预测结果。     

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.     

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。     

Numerical Method for Conditional Simulation of Levy Random Fields

Stochastic simulations of subsurface heterogeneity require accurate statistical models for spatial fluctuations. Incremental values in subsurface properties were shown previously to be approximated accurately by Levy distributions in the center and in the start of the tails of the distribution. New simulation methods utilizing these observations have been developed. Multivariate Levy distributions are used to model the multipoint joint probability density. Explicit bounds on the simulated variables prevent nonphysical extreme values and introduce a cutoff in the tails of the distribution of increments. Long-range spatial dependence is introduced through off-diagonal terms in the Levy association matrix, which is decomposed to yield a maximum likelihood type estimate at unobserved locations. This procedure reduces to a known interpolation formula developed for Gaussian fractal fields in the situation of two control points. The conditional density is not univariate Levy and is not available in closed form, but can be constructed numerically. Sequential simulation algorithms utilizing the numerically constructed conditional density successfully reproduce the desired statistical properties in simulations.     

Monitoring well placement using conditional simulation of hydraulic head

A theoretical technique for conditioning simulations in frequency domain is developed and then applied to a hydraulic-head data set from Pittman, Nevada. Frequency-domain simulation rapidly generates simulations while requiring minimal computer memory. This makes it possible by using a personal computer to create large numbers of simulations of a physical parameter field for use in studying stochastic processes. In our application, groundwater flowlines are constructed from the simulations of the hydraulic head field. Then, the crossings of the flowlines at a transect down-stream from a contaminant point source generate histograms for predicting the probability of plume interception by groundwater monitoring wells. The simulation process is discussed in detail for the Pittman study site.     

Production of conditional simulations via the LU triangular decomposition of the covariance matrix

This paper reviews the turning band method and fast Fourier transform method of producing a nonconditional simulation of a multinormal random function with a given covariance structure. A review of the two common methods of conditioning the simulation to honor the data shows that they are formally equivalent. Another method for directly pondering a conditional simulation based on the LU triangular decomposition of the covariance matrix is presented. Computational and implementation difficulties are discussed.