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。     

A multiple-point geostatistical method based on geological vector information

The reproduction of the non-stationary distribution and detailed characteristics of geological bodies is the main difficulty of reservoir modeling. Recently developed multiple-point geostatistics can represent a stationary geological body more effectively than traditional methods. When restricted to a stationary hypothesis, multiple-point geostatistical methods cannot simulate a non-stationary geological body effectively, especially when using non-stationary training images (TIs). According to geologic principles, the non-stationary distribution of geological bodies is controlled by a sedimentary model. Therefore, in this paper, we propose auxiliary variables based on the sedimentary model, namely geological vector information (GVI). GVI can characterize the non-stationary distribution of TIs and simulation domains before sequential simulation, and the precision of data event statistics will be enhanced by the sequential simulation’s data event search area limitations under the guidance of GVI. Consequently, the reproduction of non-stationary geological bodies will be improved. The key features of this method are as follows: (1) obtain TIs and geological vector information for simulated areas restricted by sedimentary models; (2) truncate TIs into a number of sub-TIs using a set of cut-off values such that each sub-TI is stationary and the adjacent sub-TIs have a certain similarity; (3) truncate the simulation domain into a number of sub-regions with the same cut-off values used in TI truncation, so that each sub-region corresponds to a number of sub-TIs; (4) use an improved method to scan the TI or TIs and construct a single search tree to restore replicates of data events located in different sub-TIs; and (5) use an improved conditional probability distribution function to perform sequential simulation. A FORTRAN program is implemented based on the SNESIM.     

Application of multiple-point geostatistics on modelling groundwater flow and transport in a cross-bedded aquifer (Belgium)

Sedimentological processes often result in complex three-dimensional subsurface heterogeneity of hydrogeological parameter values. Variogram-based stochastic approaches are often not able to describe heterogeneity in such complex geological environments. This work shows how multiple-point geostatistics can be applied in a realistic hydrogeological application to determine the impact of complex geological heterogeneity on groundwater flow and transport. The approach is applied to a real aquifer in Belgium that exhibits a complex sedimentary heterogeneity and anisotropy. A training image is constructed based on geological and hydrogeological field data. Multiple-point statistics are borrowed from this training image to simulate hydrofacies occurrence, while intrafacies permeability variability is simulated using conventional variogram-based geostatistical methods. The simulated hydraulic conductivity realizations are used as input to a groundwater flow and transport model to investigate the effect of small-scale sedimentary heterogeneity on contaminant plume migration. Results show that small-scale sedimentary heterogeneity has a significant effect on contaminant transport in the studied aquifer. The uncertainty on the spatial facies distribution and intrafacies hydraulic conductivity distribution results in a significant uncertainty on the calculated concentration distribution. Comparison with standard variogram-based techniques shows that multiple-point geostatistics allow better reproduction of irregularly shaped low-permeability clay drapes that influence solute transport.     

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.     

Direct Multiple-Point Geostatistical Simulation of Edge Properties for Modeling Thin Irregularly Shaped Surfaces

Thin, irregularly shaped surfaces such as clay drapes often have a major control on flow and transport in heterogeneous porous media. Clay drapes are often complex, curvilinear three-dimensional surfaces and display a very complex spatial distribution. Variogram-based stochastic approaches are also often not able to describe the spatial distribution of clay drapes since complex, curvilinear, continuous, and interconnected structures cannot be characterized using only two-point statistics. Multiple-point geostatistics aims to overcome the limitations of the variogram. The premise of multiple-point geostatistics is to move beyond two-point correlations between variables and to obtain (cross) correlation moments at three or more locations at a time using training images to characterize the patterns of geological heterogeneity. Multiple-point geostatistics can reproduce thin irregularly shaped surfaces such as clay drapes, but this is often computationally very intensive. This paper describes and applies a methodology to simulate thin, irregularly shaped surfaces with a smaller CPU and RAM demand than the conventional multiple-point statistical methods. The proposed method uses edge properties for indicating the presence of thin irregularly shaped surfaces. Instead of pixel values, edge properties indicating the presence of irregularly shaped surfaces are simulated using snesim. This method allows direct simulation of edge properties instead of pixel properties to make it possible to perform multiple-point geostatistical simulations with a larger cell size and thus a smaller computation time and memory demand. This method is particularly valuable for three-dimensional applications of multiple-point geostatistics.     

Comparing Training-Image Based Algorithms Using an Analysis of Distance

As additional multiple-point statistical (MPS) algorithms are developed, there is an increased need for scientific ways for comparison beyond the usual visual comparison or simple metrics, such as connectivity measures. In this paper, we start from the general observation that any (not just MPS) geostatistical simulation algorithm represents two types of variability: (1) the within-realization variability, namely, that realizations reproduce a spatial continuity model (variogram, Boolean, or training-image based), (2) the between-realization variability representing a model of spatial uncertainty. In this paper, it is argued that any comparison of algorithms needs, at a minimum, to be based on these two randomizations. In fact, for certain MPS algorithms, it is illustrated with different examples that there is often a trade-off: Increased pattern reproduction entails reduced spatial uncertainty. In this paper, the subjective choice that the best algorithm maximizes pattern reproduction is made while at the same time maximizes spatial uncertainty. The discussion is also limited to fairly standard multiple-point algorithms and that our method does not necessarily apply to more recent or possibly future developments. In order to render these fundamental principles quantitative, this paper relies on a distance-based measure for both within-realization variability (pattern reproduction) and between-realization variability (spatial uncertainty). It is illustrated in this paper that this method is efficient and effective for two-dimensional, three-dimensional, continuous, and discrete training images.     

Parallel Multiple-Point Statistics Algorithm Based on List and Tree Structures

Multiple-point statistics are widely used for the simulation of categorical variables because the method allows for integrating a conceptual model via a training image and then simulating complex heterogeneous fields. The multiple-point statistics inferred from the training image can be stored in several ways. The tree structure used in classical implementations has the advantage of being efficient in terms of CPU time, but is very RAM demanding and then implies limitations on the size of the template, which serves to make a proper reproduction of complex structures difficult. Another technique consists in storing the multiple-point statistics in lists. This alternative requires much less memory and allows for a straightforward parallel algorithm. Nevertheless, the list structure does not benefit from the shortcuts given by the branches of the tree for retrieving the multiple-point statistics. Hence, a serial algorithm based on list structure is generally slower than a tree-based algorithm. In this paper, a new approach using both list and tree structures is proposed. The idea is to index the lists by trees of reduced size: the leaves of the tree correspond to distinct sublists that constitute a partition of the entire list. The size of the indexing tree can be controlled, and then the resulting algorithm keeps memory requirements low while efficiency in terms of CPU time is significantly improved. Moreover, this new method benefits from the parallelization of the list approach.     

Adding Local Accuracy to Direct Sequential Simulation

Geostatistical simulations are globally accurate in the sense that they reproduce global statistics such as variograms and histograms. Kriging is locally accurate in the minimum local error variance sense. Building on the concept of direct sequential simulation, we propose a fast simulation method that can share these opposing objectives. It is shown that the multiple-point entropy of the resulting simulation is related to the univariate entropy of the local conditional distributions used to draw simulated values. Adding local accuracy to conditional simulations does not detract much from variogram reproduction and can be used to increase multiple-point entropy. The methods developed are illustrated using a case study.     

在储层建模中利用多点地质统计学整合地质概念模型及其解释(英文)

基于三维空间中稀疏的观测数据,地质学家和储层建模人员尝试预测井间的地质沉积相的空间非均质性时,地质概念模型和先验认识在其中扮演着重要的角色。这种整合先验模型或解释的过程有时是隐蔽或不易察觉的,正如在手工绘等值线图中的情形;它也能够被显式地运用到某种算法当中,比如数字绘图中的算法。新近兴起的多点地质统计学为地质学家和储层建模人员提供了一种有力工具,它强调使用训练图像把先验模型明确而定量地引入到储层建模当中。先验地质模型包含了被研究的真实储层中确信存在的样式,而训练图像则是该模型的定量化表达。通过再现高阶统计量,多点算法能够从训练图像中捕捉复杂的(非线性)特征样式并把它们锚定到观测的井位数据。文中描述了多点地质统计学原理,以突出训练图像概念重要性为主线,描述了多点地质统计学在建立三维储层模型中的应用。     

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.     

A Comparison of Methods for the Stochastic Simulation of Rock Fractures

Methods reported in the literature for rock fracture simulations include approaches based on stochastic geometry, multiple-point statistics and a combination of geostatistics for fracture density and object-based modelling for fracture geometries. The advantages and disadvantages of each of these approaches are discussed with examples. By way of review, the authors begin with the geostatistical indicator simulation method, based on the truncated–Gaussian algorithm; this is followed by multiple-point statistical simulation and then the stochastic geometry approach, which is based on marked point process simulation. A new approach, based on pluriGaussian structural simulation, is then introduced. The new approach incorporates in the simulation the spatial correlation between different sets of fractures, which in general, is very difficult, if not impossible, to accomplish in the three methods reviewed. Each simulation method is summarised together with detailed simulation procedures for each. A published two-dimensional fracture dataset is used as a means of assessing the performance of each simulation method and of demonstrating the concepts discussed in the text.     

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.     

A Comparison of Random Field Models Beyond Bivariate Distributions

In order to determine to what extent a spatial random field can be characterized by its low-order distributions, we consider four models (specifically, random spatial tessellations) with exactly the same univariate and bivariate distributions and we compare the statistics associated with various multiple-point configurations and the responses to specific transfer functions. The three- and four-point statistics are found to be the same or experimentally hardly distinguishable because of ergodic fluctuations, whereas change of support and flow simulation produce very different outcomes. This example indicates that low-order distributions may not discriminate between contending random field models, that simulation algorithms based on such distributions may not reproduce the spatial properties of a given model or training image, and that the inference of high-order distribution may require very large training images.     

多点地质统计学建模的发展趋势

从算法研究、训练图像处理和实际应用三个方面详细解剖了国内外多点地质统计学的发展历程,在此基础上,分析了多点地质统计学主流的几种算法的核心原理、适用范围及优缺点,以此来对储层建模的发展趋势作出展望。目前,多点地质统计学虽是随机建模的一种前沿研究热点,但由于其尚未成熟,仍需对建模算法进行研究。为此,在前人研究的基础上,重点分析了多点地质统计学的发展趋势：合理处理训练图像;合理利用软信息;选择合适的相似性方法;选择合适的标准化方法;合理利用平稳性;算法间的耦合;选择合适的过滤器;拓展缝洞型碳酸盐岩模拟。最后,提出多点地质统计学在储层建模方面,应从增加储层的模拟区域、提高模拟精度、扩大储层相的模拟范围和提高计算机模拟效率等方面进行改进。     

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.     

High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

The spatial distributions of earth science and engineering phenomena under study are currently predicted from finite measurements and second-order geostatistical models. The latter models can be limiting, as geological systems are highly complex, non-Gaussian, and exhibit non-linear patterns of spatial connectivity. Non-linear and non-Gaussian high-order geostatistics based on spatial connectivity measures, namely spatial cumulants, are proposed as a new alternative modeling framework for spatial data. This framework has two parts. The first part is the definition, properties, and inference of spatial cumulants—including understanding the interrelation of cumulant characteristics with the in-situ behavior of geological entities or processes, as examined in this paper. The second part is the research on a random field model for simulation based on its high-order spatial cumulants. Mathematical definitions of non-Gaussian spatial random functions and their high-order spatial statistics are presented herein, stressing the notion of spatial cumulants. The calculation of spatial cumulants with spatial templates follows, including anisotropic experimental cumulants. Several examples of two- and three-dimensional images, including a diamond bearing kimberlite pipe from the Ekati Mine in Canada, are analyzed to assess the relations between cumulants and the spatial behavior of geological processes. Spatial cumulants of orders three to five are shown to capture directional multiple-point periodicity, connectivity including connectivity of extreme values, and spatial architecture. In addition, they provide substantial information on geometric characteristics and anisotropy of geological patterns. It is further shown that effects of complex spatial patterns are seen even if only subsets of all cumulant templates are computed. Compared to second-order statistics, cumulant maps are found to include a wealth of additional information from underlying geological patterns. Further work seeks to integrate this information in the predictive capabilities of a random field model.     

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.     

Indicator Simulation Accounting for Multiple-Point Statistics

Geostatistical simulation aims at reproducing the variability of the real underlying phenomena. When nonlinear features or large-range connectivity is present, the traditional variogram-based simulation approaches do not provide good reproduction of those features. Connectivity of high and low values is often critical for grades in a mineral deposit. Multiple-point statistics can help to characterize these features. The use of multiple-point statistics in geostatistical simulation was proposed more than 10 years ago, on the basis of the use of training images to extract the statistics. This paper proposes the use of multiple-point statistics extracted from actual data. A method is developed to simulate continuous variables. The indicator kriging probabilities used in sequential indicator simulation are modified by probabilities extracted from multiple-point configurations. The correction is done under the assumption of conditional independence. The practical implementation of the method is illustrated with data from a porphyry copper mine.     

Solving Speed and Memory Issues in Multiple-Point Statistics Simulation Program SNESIM

In the last 10 years, Multiple-Point Statistics (MPS) modeling has emerged in Geostatistics as a valuable alternative to traditional variogram-based and object-based modeling. In contrast to variogram-based simulation, which is limited to two-point correlation reproduction, MPS simulation extracts and reproduces multiple-point statistics moments from training images; this allows modeling geologically realistic features, such as channels that control reservoir connectivity and flow behavior. In addition, MPS simulation works on individual pixels or small groups of pixels (patterns), thus does not suffer from the same data conditioning limitations as object-based simulation. The Single Normal Equation Simulation program SNESIM was the first implementation of MPS simulation to propose, through the introduction of search trees, an efficient solution to the extraction and storage of multiple-point statistics moments from training images. SNESIM is able to simulate three-dimensional models; however, memory and speed issues can occur when applying it to multimillion cell grids. Several other implementations of MPS simulation were proposed after SNESIM, but most of them manage to reduce memory demand or simulation time only at the expense of data conditioning exactitude and/or training pattern reproduction quality. In this paper, the original SNESIM program is revisited, and solutions are presented to eliminate both memory demand and simulation time limitations. First, we demonstrate that the time needed to simulate a grid node is a direct function of the number of uninformed locations in the conditioning data search neighborhood. Thus, two improvements are proposed to maximize the ratio of informed to uniformed locations in search neighborhoods: a new multiple-grid approach introducing additional intermediary subgrids; and a new search neighborhood designing process to preferentially include previously simulated node locations. Finally, because SNESIM memory demand and simulation time increase with the size of the data template used to extract multiple-point statistics moments from the training image and build the search tree, a simple method is described to minimize data template sizes while preserving training pattern reproduction quality.