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.     

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...     

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.     

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.     

Simulation of Geological Contacts from Interpreted Geological Model Using Multiple-Point Statistics

Applications of multiple-point statistics (mps) algorithms to large non-repetitive geological objects such as those found in mining deposits are difficult because most mps algorithms rely on pattern repetition for simulation. In many cases, an interpreted geological model built from a computer-aided design system is readily available but suffers as a training image due to the lack of patterns repetitiveness. Porphyry copper deposits and iron ore formations are good examples of such mining deposits with non-repetitive patterns. This paper presents an algorithm called contactsim that focuses on reproducing the patterns of the contacts between geological types. The algorithm learns the shapes of the lithotype contacts as interpreted by the geologist, and simulates their patterns at a later stage. Defining a zone of uncertainty around the lithological contact is a critical step in contactsim, because it defines both the zones where the simulation is performed and where the algorithm should focus to learn the transitional patterns between lithotypes. A larger zone of uncertainty results in greater variation between realizations. The definition of the uncertainty zone must take into consideration the geological understanding of the deposit, and the reliability of the contact zones. The contactsim algorithm is demonstrated on an iron ore formation.     

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.     

Multiple-Point Statistics for Modeling Facies Heterogeneities in a Porous Medium: The Komadugu-Yobe Alluvium,Lake Chad Basin

Multiple-point statistics are used to model facies heterogeneities in the vadose zone of the Komadugu-Yobe River valley (southeastern Niger) which is presently submitted to an undergoing intensive agricultural development; therefore, increasing quantitative and qualitative pressures are exerted on groundwater resources. The sand–clay heterogeneities are analyzed by means of a Landsat image acquired during a high flow period over a 160 km stretch in the downstream part of the valley and a set of 50 boreholes drilled near the town of Diffa (4 km×4 km area). The horizontal variograms of heterogeneities are characterized by a noticeably constant length scale of 380 m and clayey objects are shown to be randomly distributed in space according to a Poisson process. A set of two-dimensional vertical images is built based on a Boolean procedure and the Snesim algorithm is used to simulate synthetic three-dimensional media. When the vertical correlation length is fitted, the three-dimensional model satisfactorily reproduces the second order statistics of heterogeneities and the specific facies patterns.     

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.     

Sampling Strategies for Uncertainty Reduction in Categorical Random Fields: Formulation,Mathematical Analysis and Application to Multiple-Point Simulations

Mathematical Geosciences - The task of optimal sampling for the statistical simulation of a discrete random field is addressed from the perspective of minimizing the posterior uncertainty of...     

History Matching Geostatistical Model Realizations Using a Geometrical Domain Based Parameterization Technique

Reservoir characterization needs the integration of various data through history matching, especially dynamic information such as production or 4D seismic data. Although reservoir heterogeneities are commonly generated using geostatistical models, random realizations cannot generally match observed dynamic data. To constrain model realizations to reproduce measured dynamic data, an optimization procedure may be applied in an attempt to minimize an objective function, which quantifies the mismatch between real and simulated data. Such assisted history matching methods require a parameterization of the geostatistical model to allow the updating of an initial model realization. However, there are only a few parameterization methods available to update geostatistical models in a way consistent with the underlying geostatistical properties. This paper presents a local domain parameterization technique that updates geostatistical realizations using assisted history matching. This technique allows us to locally change model realizations through the variation of geometrical domains whose geometry and size can be easily controlled and parameterized. This approach provides a new way to parameterize geostatistical realizations in order to improve history matching efficiency.     

A Deep-Learning-Based Geological Parameterization for History Matching Complex Models

A new low-dimensional parameterization based on principal component analysis (PCA) and convolutional neural networks (CNN) is developed to represent complex geological models. The CNN–PCA method is inspired by recent developments in computer vision using deep learning. CNN–PCA can be viewed as a generalization of an existing optimization-based PCA (O-PCA) method. Both CNN–PCA and O-PCA entail post-processing a PCA model to better honor complex geological features. In CNN–PCA, rather than use a histogram-based regularization as in O-PCA, a new regularization involving a set of metrics for multipoint statistics is introduced. The metrics are based on summary statistics of the nonlinear filter responses of geological models to a pre-trained deep CNN. In addition, in the CNN–PCA formulation presented here, a convolutional neural network is trained as an explicit transform function that can post-process PCA models quickly. CNN–PCA is shown to provide both unconditional and conditional realizations that honor the geological features present in reference SGeMS geostatistical realizations for a binary channelized system. Flow statistics obtained through simulation of random CNN–PCA models closely match results for random SGeMS models for a demanding case in which O-PCA models lead to significant discrepancies. Results for history matching are also presented. In this assessment CNN–PCA is applied with derivative-free optimization, and a subspace randomized maximum likelihood method is used to provide multiple posterior models. Data assimilation and significant uncertainty reduction are achieved for existing wells, and physically reasonable predictions are also obtained for new wells. Finally, the CNN–PCA method is extended to a more complex nonstationary bimodal deltaic fan system, and is shown to provide high-quality realizations for this challenging example.

     

Bayesian Reservoir History Matching Considering Model and Parameter Uncertainties

This paper presents a consistent Bayesian solution for data integration and history matching for oil reservoirs while accounting for both model and parameter uncertainties. The developed method uses Gaussian Process Regression to build a permeability map conforming to collected data at well bores. Following that, an augmented Markov Chain Monte Carlo sampler is used to condition the permeability map to dynamic production data. The selected proposal distribution for the Markov Chain Monte Carlo conforms to the Gaussian process regression output. The augmented Markov Chain Monte Carlo sampler allows transition steps between different models of the covariance function, and hence both the parameter and model space are effectively explored. In contrast to single model Markov Chain Monte Carlo samplers, the proposed augmented Markov Chain Monte Carlo sampler eliminates the selection bias of certain covariance structures of the inferred permeability field. The proposed algorithm can be used to account for general model and parameter uncertainties.     

Hybrid Iterative Ensemble Smoother for History Matching of Hierarchical Models

Mathematical Geosciences - The choice of a prior model can have a large impact on the ability to assimilate data. In standard applications of ensemble-based data assimilation, all realizations in...     

多点地质统计学在河流相储层建模中的应用

多点地质统计学综合了基于象元方法以及基于目标方法两者的优点,对于河流相等具有复杂地质形态的储层精确建模具有较强的优势.在对传统建模方法综合分析的基础上,介绍了多点地质统计学的基本理论及SNESIM算法,并应用该技术对大牛地气田某开发井区的辫状分流河道相进行了实际建模.研究结果表明,在河流相储层建模中,该方法比传统的建模方法更具优越性.最后,进一步综合讨论了多点地质统计学目前面临的主要问题(包括训练图像、目标体连续性、数据样板选择、综合地震信息等方面)的改进方法.     

Determination of Two-phase Relative Permeability from a Displacement with Safman-Rayleigh Instability Using a Coarse-Scale Model History Matching Approach

Computational Geosciences - In heavy oil displacement by fluid injection, severe instability can occur due to the adverse mobility ratio, gravity segregation or compositional effects. However, when...     

Numerical ANFIS-Based Formulation for Prediction of the Ultimate Axial Load Bearing Capacity of Piles Through CPT Data

This study explores the potential of adaptive neuro-fuzzy inference systems (ANFIS) for prediction of the ultimate axial load bearing capacity of piles (P_u) using cone penetration test (CPT) data. In this regard, a reliable previously published database composed of 108 datasets was selected to develop ANFIS models. The collected database contains information regarding pile geometry, material, installation, full-scale static pile load test and CPT results for each sample. Reviewing the literature, several common and uncommon variables have been considered for direct or indirect estimation of P_u based on static pile load test, cone penetration test data or other in situ or laboratory testing methods. In present study, the pile shaft and tip area, the average cone tip resistance along the embedded length of the pile, the average cone tip resistance over influence zone and the average sleeve friction along the embedded length of the pile which are obtained from CPT data are considered as independent input variables where the output variable is P_u for the ANFIS model development. Besides, a notable criticism about ANFIS as a prediction tool is that it does not provide practical prediction equations. To tackle this issue, the obtained optimal ANFIS model is represented as a tractable equation which can be used via spread sheet software or hand calculations to provide precise predictions of P_u with the calculated correlation coefficient of 0.96 between predicted and experimental values for all of the data in this study. Considering several criteria, it is represented that the proposed model is able to estimate the output with a high degree of accuracy as compared to those results obtained by some direct CPT-based methods in the literature. Furthermore, in order to assess the capability of the proposed model from geotechnical engineering viewpoints, sensitivity and parametric analyses are done.     

Adaptive Conditioning of Multiple-Point Statistical Facies Simulation to Flow Data with Probability Maps

Multiple-point statistics (MPS) provides a flexible grid-based approach for simulating complex geologic patterns that contain high-order statistical information represented by a conceptual prior geologic model known as a training image (TI). While MPS is quite powerful for describing complex geologic facies connectivity, conditioning the simulation results on flow measurements that have a nonlinear and complex relation with the facies distribution is quite challenging. Here, an adaptive flow-conditioning method is proposed that uses a flow-data feedback mechanism to simulate facies models from a prior TI. The adaptive conditioning is implemented as a stochastic optimization algorithm that involves an initial exploration stage to find the promising regions of the search space, followed by a more focused search of the identified regions in the second stage. To guide the search strategy, a facies probability map that summarizes the common features of the accepted models in previous iterations is constructed to provide conditioning information about facies occurrence in each grid block. The constructed facies probability map is then incorporated as soft data into the single normal equation simulation (snesim) algorithm to generate a new candidate solution for the next iteration. As the optimization iterations progress, the initial facies probability map is gradually updated using the most recently accepted iterate. This conditioning process can be interpreted as a stochastic optimization algorithm with memory where the new models are proposed based on the history of the successful past iterations. The application of this adaptive conditioning approach is extended to the case where multiple training images are proposed as alternative geologic scenarios. The advantages and limitations of the proposed adaptive conditioning scheme are discussed and numerical experiments from fluvial channel formations are used to compare its performance with non-adaptive conditioning techniques.     

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.