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.     

Fast FILTERSIM Simulation with Score-based Distance

FILTERSIM is a pattern-based multiple-point geostatistical algorithm for modeling both continuous and categorical variables. It first groups all the patterns from a training image into a set of pattern classes using their filter scores. At each simulation location, FILTERSIM identifies the training pattern class closest to the local conditioning data event, then samples a training pattern from that prototype class and pastes it onto the simulation grid. In the original FILTERSIM algorithm, the selection of the closest pattern class is based on the pixel-wise distance between the prototype of each training pattern class and the local conditioning data event. Hence, FILTERSIM is computationally intensive for 3D simulations, especially with a large and pattern-rich training image. In this paper, a novel approach is proposed to accelerate the simulation process by replacing that pixel-wise distance calculation with a filter score comparison, which is the difference between the filter score of local conditioning data event and that of each pattern prototype. This score-based distance calculation significantly reduces the CPU consumption due to the tremendous data dimension reduction. The results show that this new score based-distance calculation can speed up FILTERSIM simulation by a factor up to 10 in 3D applications.     

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.     

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。     

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.     

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.     

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.     

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.     

Stationarity Scores on Training Images for Multipoint Geostatistics

This research introduces a novel method to assess the validity of training images used as an input for Multipoint Geostatistics, alternatively called Multiple Point Simulation (MPS). MPS are a family of spatial statistical interpolation algorithms that are used to generate conditional simulations of property fields such as geological facies. They are able to honor absolute “hard” constraints (e.g., borehole data) as well as “soft” constraints (e.g., probability fields derived from seismic data, and rotation and scale). These algorithms require 2D or 3D training images or analogs whose textures represent a spatial arrangement of geological properties that is presumed to be similar to that of a target volume to be modeled. To use the current generation of MPS algorithms, statistically valid training image are required as input. In this context, “statistical validity” includes a requirement of stationarity, so that one can derive from the training image an average template pattern. This research focuses on a practical method to assess stationarity requirements for MPS algorithms, i.e., that statistical density or probability distribution of the quantity shown on the image does not change spatially, and that the image shows repetitive shapes whose orientation and scale are spatially constant. This method employs image-processing techniques based on measures of stationarity of the category distribution, the directional (or orientation) property field and the scale property field of those images. It was successfully tested on a set of two-dimensional images representing geological features and its predictions were compared to actual realizations of MPS algorithms. An extension of the algorithms to 3D images is also proposed. As MPS algorithms are being used increasingly in hydrocarbon reservoir modeling, the methods described should facilitate screening and selection of the input training images.     

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.     

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.     

Teacher''s Aide Variogram Interpretation and Modeling

The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology.     

Teacher's Aide Variogram Interpretation and Modeling

The variogram is a critical input to geostatistical studies: (1) it is a tool to investigate and quantify the spatial variability of the phenomenon under study, and (2) most geostatistical estimation or simulation algorithms require an analytical variogram model, which they will reproduce with statistical fluctuations. In the construction of numerical models, the variogram reflects some of our understanding of the geometry and continuity of the variable, and can have a very important impact on predictions from such numerical models. The principles of variogram modeling are developed and illustrated with a number of practical examples. A three-dimensional interpretation of the variogram is necessary to fully describe geologic continuity. Directional continuity must be described simultaneously to be consistent with principles of geological deposition and for a legitimate measure of spatial variability for geostatistical modeling algorithms. Interpretation principles are discussed in detail. Variograms are modeled with particular functions for reasons of mathematical consistency. Used correctly, such variogram models account for the experimental data, geological interpretation, and analogue information. The steps in this essential data integration exercise are described in detail through the introduction of a rigorous methodology.     

Multiple-point geostatistical modeling based on the cross-correlation functions

An important issue in reservoir modeling is accurate generation of complex structures. The problem is difficult because the connectivity of the flow paths must be preserved. Multiple-point geostatistics is one of the most effective methods that can model the spatial patterns of geological structures, which is based on an informative geological training image that contains the variability, connectivity, and structural properties of a reservoir. Several pixel- and pattern-based methods have been developed in the past. In particular, pattern-based algorithms have become popular due to their ability for honoring the connectivity and geological features of a reservoir. But a shortcoming of such methods is that they require a massive data base, which make them highly memory- and CPU-intensive. In this paper, we propose a novel methodology for which there is no need to construct pattern data base and small data event. A new function for the similarity of the generated pattern and the training image, based on a cross-correlation (CC) function, is proposed that can be used with both categorical and continuous training images. We combine the CC function with an overlap strategy and a new approach, adaptive recursive template splitting along a raster path, in order to develop an algorithm, which we call cross-correlation simulation (CCSIM), for generation of the realizations of a reservoir with accurate conditioning and continuity. The performance of CCSIM is tested for a variety of training images. The results, when compared with those of the previous methods, indicate significant improvement in the CPU and memory requirements.     

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.     

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。     

Facies Modeling Using a Markov Mesh Model Specification

The spatial continuity of facies is one of the key factors controlling flow in reservoir models. Traditional pixel-based methods such as truncated Gaussian random fields and indicator simulation are based on only two-point statistics, which is insufficient to capture complex facies structures. Current methods for multi-point statistics either lack a consistent statistical model specification or are too computer intensive to be applicable. We propose a Markov mesh model based on generalized linear models for geological facies modeling. The approach defines a consistent statistical model that is facilitated by efficient estimation of model parameters and generation of realizations. Our presentation includes a formulation of the general framework, model specifications in two and three dimensions, and details on how the parameters can be estimated from a training image. We illustrate the method using multiple training images, including binary and trinary images and simulations in two and three dimensions. We also do a thorough comparison to the snesim approach. We find that the current model formulation is applicable for multiple training images and compares favorably to the snesim approach in our test examples. The method is highly memory efficient.     

A flow-based pattern recognition algorithm for rapid quantification of geologic uncertainty

Geologic uncertainties and limited well data often render recovery forecasting a difficult undertaking in typical appraisal and early development settings. Recent advances in geologic modeling algorithms permit automation of the model generation process via macros and geostatistical tools. This allows rapid construction of multiple alternative geologic realizations. Despite the advances in geologic modeling, computation of the reservoir dynamic response via full-physics reservoir simulation remains a computationally expensive task. Therefore, only a few of the many probable realizations are simulated in practice. Experimental design techniques typically focus on a few discrete geologic realizations as they are inherently more suitable for continuous engineering parameters and can only crudely approximate the impact of geology. A flow-based pattern recognition algorithm (FPRA) has been developed for quantifying the forecast uncertainty as an alternative. The proposed algorithm relies on the rapid characterization of the geologic uncertainty space represented by an ensemble of sufficiently diverse static model realizations. FPRA characterizes the geologic uncertainty space by calculating connectivity distances, which quantify how different each individual realization is from all others in terms of recovery response. Fast streamline simulations are employed in evaluating these distances. By applying pattern recognition techniques to connectivity distances, a few representative realizations are identified within the model ensemble for full-physics simulation. In turn, the recovery factor probability distribution is derived from these intelligently selected simulation runs. Here, FPRA is tested on an example case where the objective is to accurately compute the recovery factor statistics as a function of geologic uncertainty in a channelized turbidite reservoir. Recovery factor cumulative distribution functions computed by FPRA compare well to the one computed via exhaustive full-physics simulations.     

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.