Can a Training Image Be a Substitute for a Random Field Model?

In most multiple-point simulation algorithms, all statistical features are provided by one or several training images (TI) that serve as a substitute for a random field model. However, because in practice the TI is always of finite size, the stochastic nature of multiple-point simulation is questionable. This issue is addressed by considering the case of a sequential simulation algorithm applied to a binary TI that is a genuine realization of an underlying random field. At each step, the algorithm uses templates containing the current target point as well as all previously simulated points. The simulation is validated by checking that all statistical features of the random field (supported by the simulation domain) are retrieved as an average over a large number of outcomes. The results are as follows. It is demonstrated that multiple-point simulation performs well whenever the TI is a complete (infinitely large) realization of a stationary, ergodic random field. As soon as the TI is restricted to a limited domain, the statistical features cannot be obtained exactly, but integral range techniques make it possible to predict how much the TI should be extended to approximate them up to a prespecified precision. Moreover, one can take advantage of extending the TI to reduce the number of disruptions in the execution of the algorithm, which arise when no conditioning template can be found in the TI.     

Modelling a real-world buried valley system with vertical non-stationarity using multiple-point statistics

Stationarity has traditionally been a requirement of geostatistical simulations. A common way to deal with non-stationarity is to divide the system into stationary sub-regions and subsequently merge the realizations for each region. Recently, the so-called partition approach that has the flexibility to model non-stationary systems directly was developed for multiple-point statistics simulation (MPS). The objective of this study is to apply the MPS partition method with conventional borehole logs and high-resolution airborne electromagnetic (AEM) data, for simulation of a real-world non-stationary geological system characterized by a network of connected buried valleys that incise deeply into layered Miocene sediments (case study in Denmark). The results show that, based on fragmented information of the formation boundaries, the MPS partition method is able to simulate a non-stationary system including valley structures embedded in a layered Miocene sequence in a single run. Besides, statistical information retrieved from the AEM data improved the simulation of the geology significantly, especially for the deep-seated buried valley sediments where borehole information is sparse.     

Validation Techniques for Geological Patterns Simulations Based on Variogram and Multiple-Point Statistics

Traditional simulation methods that are based on some form of kriging are not sensitive to the presence of strings of connectivity of low or high values. They are particularly inappropriate in many earth sciences applications, where the geological structures to be simulated are curvilinear. In such cases, techniques allowing the reproduction of multiple-point statistics are required. The aim of this paper is to point out the advantages of integrating such multiple-statistics in a model in order to allow shape reproduction, as well as heterogeneity structures, of complex geological patterns to emerge. A comparison between a traditional variogram-based simulation algorithm, such as the sequential indicator simulation, and a multiple-point statistics algorithm (e.g., the single normal equation simulation) is presented. In particular, it is shown that the spatial distribution of limestone with meandering channels in Lecce, Italy is better reproduced by using the latter algorithm. The strengths of this study are, first, the use of a training image that is not a fluvial system and, more importantly, the quantitative comparison between the two algorithms. The paper focuses on different metrics that facilitate the comparison of the methods used for limestone spatial distribution simulation: both objective measures of similarity of facies realizations and high-order spatial cumulants based on different third- and fourth-order spatial templates are considered.     

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.     

Application of Multiple Point Geostatistics to Non-stationary Images

Simulation of flow and solute transport through aquifers or oil reservoirs requires a precise representation of subsurface heterogeneity that can be achieved by stochastic simulation approaches. Traditional geostatistical methods based on variograms, such as truncated Gaussian simulation or sequential indicator simulation, may fail to generate the complex, curvilinear, continuous and interconnected facies distributions that are often encountered in real geological media, due to their reliance on two-point statistics. Multiple Point Geostatistics (MPG) overcomes this constraint by using more complex point configurations whose statistics are retrieved from training images. Obtaining representative statistics requires stationary training images, but geological understanding often suggests a priori facies variability patterns. This research aims at extending MPG to non-stationary facies distributions. The proposed method subdivides the training images into different areas. The statistics for each area are stored in separate frequency search trees. Several training images are used to ensure that the obtained statistics are representative. The facies probability distribution for each cell during simulation is calculated by weighting the probabilities from the frequency trees. The method is tested on two different object-based training image sets. Results show that non-stationary training images can be used to generate suitable non-stationary facies distributions.     

Using Multiple-Point Geostatistics for Tracer Test Modeling in a Clay-Drape Environment with Spatially Variable Conductivity and Sorption Coefficient

This study investigates the effect of fine-scale clay drapes on tracer transport. A tracer test was performed in a sandbar deposit consisting of cross-bedded sandy units intercalated with many fine-scale clay drapes. The heterogeneous spatial distribution of the clay drapes causes a spatially variable hydraulic conductivity and sorption coefficient. A fluorescent tracer (sodium naphthionate) was injected in two injection wells and ground water was sampled and analyzed from five pumping wells. To determine (1) whether the fine-scale clay drapes have a significant effect on the measured concentrations and (2) whether application of multiple-point geostatistics can improve interpretation of tracer tests in media with complex geological heterogeneity, this tracer test is analyzed with a local three-dimensional ground-water flow and transport model in which fine-scale sedimentary heterogeneity is modeled using multiple-point geostatistics. To reduce memory needs and calculation time for the multiple-point geostatistical simulation step, this study uses the technique of direct multiple-point geostatistical simulation of edge properties. Instead of simulating pixel values, model cell edge properties indicating the presence of irregularly shaped surfaces are simulated using multiple-point geostatistical simulations. Results of a sensitivity analysis show under which conditions clay drapes have a significant effect on the concentration distribution. Calibration of the model against measured concentrations from the tracer tests reduces the uncertainty on the clay-drape parameters. The calibrated model shows which features of the breakthrough curves can be attributed to the geological heterogeneity of the aquifer and which features are caused by other processes.     

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.     

Fast direct sampling for multiple-point stochastic simulation

Multiple-point statistics simulation has recently attracted significant attention for the simulation of complex geological structures. In this paper, a fast direct sampling (FDS) algorithm is presented based on a fast gradient descent pattern matching strategy. The match is directly extracted from the training image (TI) and so the method does not require intensive preprocessing and database storage. The initial node of the search path is selected randomly but the following nodes are selected in a principled manner so that the path is conducted to the right match. Each node is selected based on the matching accuracy and the behavior of the TI in the previous node. A simple initialization strategy is presented in this paper which significantly accelerates the matching process at the expense of a very naïve preprocessing stage. The proposed simulation algorithm has several outstanding advantages: it needs no (or very limited) preprocessing, does not need any database storage, searches for the match directly in the TI, is not limited to fixed size patterns (the pattern size can be easily changed during simulation), is capable of handling both continuous and categorical data, is capable of handling multivariate data, and finally and more importantly, is a fast method while maintaining high standards for the matching quality. Experiments on different TIs reveal that the simulation results of FDS and DS are comparable in terms of pattern reproduction and connectivity while FDS is far faster than DS.     

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 Pattern-Based Simulation of Non-stationary Geostatistical Models

Non-stationary models often capture better spatial variation of real world spatial phenomena than stationary ones. However, the construction of such models can be tedious as it requires modeling both statistical trend and stationary stochastic component. Non-stationary models are an important issue in the recent development of multiple-point geostatistical models. This new modeling paradigm, with its reliance on the training image as the source for spatial statistics or patterns, has had considerable practical appeal. However, the role and construction of the training image in the non-stationary case remains a problematic issue from both a modeling and practical point of view. In this paper, we provide an easy to use, computationally efficient methodology for creating non-stationary multiple-point geostatistical models, for both discrete and continuous variables, based on a distance-based modeling and simulation of patterns. In that regard, the paper builds on pattern-based modeling previously published by the authors, whereby a geostatistical realization is created by laying down patterns as puzzle pieces on the simulation grid, such that the simulated patterns are consistent (in terms of a similarity definition) with any previously simulated ones. In this paper we add the spatial coordinate to the pattern similarity calculation, thereby only borrowing patterns locally from the training image instead of globally. The latter would entail a stationary assumption. Two ways of adding the geographical coordinate are presented, (1) based on a functional that decreases gradually away from the location where the pattern is simulated and (2) based on an automatic segmentation of the training image into stationary regions. Using ample two-dimensional and three-dimensional case studies we study the behavior in terms of spatial and ensemble uncertainty of the generated realizations.     

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 Training Image Approach for Spatial Modeling of Geologic Domains

Characterization of complex geological features and patterns remains one of the most challenging tasks in geostatistics. Multiple point statistics (MPS) simulation offers an alternative to accomplish this aim by going beyond classical two-point statistics. Reproduction of features in the final realizations is achieved by borrowing high-order spatial statistics from a training image. Most MPS algorithms use one training image at a time chosen by the geomodeler. This paper proposes the use of multiple training images simultaneously for spatial modeling through a scheme of data integration for conditional probabilities known as a linear opinion pool. The training images (TIs) are based on the available information and not on conceptual geological models; one image comes from modeling the categories by a deterministic approach and another comes from the application of conventional sequential indicator simulation. The first is too continuous and the second too random. The mixing of TIs requires weights for each of them. A methodology for calibrating the weights based on the available drillholes is proposed. A measure of multipoint entropy along the drillholes is matched by the combination of the two TIs. The proposed methodology reproduces geologic features from both TIs with the correct amount of continuity and variability. There is no need for a conceptual training image from another modeling technique; the data-driven TIs permit a robust inference of spatial structure from reasonably spaced drillhole data.     

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.     

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.     

Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling

The advent of multiple-point geostatistics (MPS) gave rise to the integration of complex subsurface geological structures and features into the model by the concept of training images. Initial algorithms generate geologically realistic realizations by using these training images to obtain conditional probabilities needed in a stochastic simulation framework. More recent pattern-based geostatistical algorithms attempt to improve the accuracy of the training image pattern reproduction. In these approaches, the training image is used to construct a pattern database. Consequently, sequential simulation will be carried out by selecting a pattern from the database and pasting it onto the simulation grid. One of the shortcomings of the present algorithms is the lack of a unifying framework for classifying and modeling the patterns from the training image. In this paper, an entirely different approach will be taken toward geostatistical modeling. A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge will be presented.     

Geostatistical Seismic Inversion with Direct Sequential Simulation and Co-simulation with Multi-local Distribution Functions

Stochastic sequential simulation is a common modelling technique used in Earth sciences and an integral part of iterative geostatistical seismic inversion methodologies. Traditional stochastic sequential simulation techniques based on bi-point statistics assume, for the entire study area, stationarity of the spatial continuity pattern and a single probability distribution function, as revealed by a single variogram model and inferred from the available experimental data, respectively. In this paper, the traditional direct sequential simulation algorithm is extended to handle non-stationary natural phenomena. The proposed stochastic sequential simulation algorithm can take into consideration multiple regionalized spatial continuity patterns and probability distribution functions, depending on the spatial location of the grid node to be simulated. This work shows the application and discusses the benefits of the proposed stochastic sequential simulation as part of an iterative geostatistical seismic inversion methodology in two distinct geological environments in which non-stationarity behaviour can be assessed by the simultaneous interpretation of the available well-log and seismic reflection data. The results show that the elastic models generated by the proposed stochastic sequential simulation are able to reproduce simultaneously the regional and global variogram models and target distribution functions relative to the average volume of each sub-region. When used as part of a geostatistical seismic inversion procedure, the retrieved inverse models are more geologically realistic, since they incorporate the knowledge of the subsurface geology as provided, for example, by seismic and well-log data interpretation.     

塔河油气田AT1区块凝析气藏三维地质建模研究

以随机函数理论为基础,采用相控-多参数协同的随机建模方法,建立塔河油气田AT1区块凝析气藏三维地质模型,实现气藏精细三维表征。首先,以钻井和岩芯资料为基础构建储层构造模型;然后,以小层界面为控制条件建立储层结构模型;接着,在沉积相、地质条件的约束下,采用序贯指示模拟法来建立砂体骨架模型;随后,在砂体骨架模型内进行优势相计算,形成最终有效砂体骨架模型;最后,以有效砂体骨架模型为约束,采用序贯高斯模拟法建立储层物性参数模型。结果表明：将物性参数变量与微相分布结合的序贯高斯模拟法建立孔隙度等物性参数的分布模型,以及采用地质分析类比、地质统计分析等方法优选最佳模型是有效的地质建模方法;所建地质模型精确细致地表征了塔河油气田AT1区块凝析气藏构造格架及储层、流体三维分布,反映了辫状水道复合连片,东北向展布,储层物性受相控较明显,非均质性较强。     

多点地质统计学在苏里格气田地质建模的应用

多点地质统计学是一种建立地质模型的统计学方法,该方法比传统的两点地质统计学更适合河流相沉积体系。本文综合地震、测井、录井及生产资料,绘制了长庆油田苏里格气田苏48区块盒8下段辫状河训练图像,并在此基础上利用多点地质统计学方法,加入三维地震资料作为约束,以水平井整体开发为研究对象,建立了该区的地质模型,优化了水平井整体部署,指导水平井导向。利用多口水平井实钻效果验证了地质模型的精度,以苏19-62井为例得到:录井显示气层钻遇率为78.3%,测井显示气层钻遇率为69%。这与地震反演剖面预测相似,且与实钻效果有很好的对应性。     

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.     

Multiple-Point Simulation with an Existing Reservoir Model as Training Image

The multiple-point simulation (MPS) method has been increasingly used to describe the complex geologic features of petroleum reservoirs. The MPS method is based on multiple-point statistics from training images that represent geologic patterns of the reservoir heterogeneity. The traditional MPS algorithm, however, requires the training images to be stationary in space, although the spatial distribution of geologic patterns/features is usually nonstationary. Building geologically realistic but statistically stationary training images is somehow contradictory for reservoir modelers. In recent research on MPS, the concept of a training image has been widely extended. The MPS approach is no longer restricted by the size or the stationarity of training images; a training image can be a small geometrical element or a full-field reservoir model. In this paper, the different types of training images and their corresponding MPS algorithms are first reviewed. Then focus is placed on a case where a reservoir model exists, but needs to be conditioned to well data. The existing model can be built by process-based, object-based, or any other type of reservoir modeling approach. In general, the geologic patterns in a reservoir model are constrained by depositional environment, seismic data, or other trend maps. Thus, they are nonstationary, in the sense that they are location dependent. A new MPS algorithm is proposed that can use any existing model as training image and condition it to well data. In particular, this algorithm is a practical solution for conditioning geologic-process-based reservoir models to well data.