A probabilistic parametrization for geological uncertainty estimation using the ensemble Kalman filter (EnKF)

In the past years, many applications of history-matching methods in general and ensemble Kalman filter in particular have been proposed, especially in order to estimate fields that provide uncertainty in the stochastic process defined by the dynamical system of hydrocarbon recovery. Such fields can be permeability fields or porosity fields, but can also fields defined by the rock type (facies fields). The estimation of the boundaries of the geologic facies with ensemble Kalman filter (EnKF) was made, in different papers, with the aid of Gaussian random fields, which were truncated using various schemes and introduced in a history-matching process. In this paper, we estimate, in the frame of the EnKF process, the locations of three facies types that occur into a reservoir domain, with the property that each two could have a contact. The geological simulation model is a form of the general truncated plurigaussian method. The difference with other approaches consists in how the truncation scheme is introduced and in the observation operator of the facies types at the well locations. The projection from the continuous space of the Gaussian fields into the discrete space of the facies fields is realized through in an intermediary space (space with probabilities). This space connects the observation operator of the facies types at the well locations with the geological simulation model. We will test the model using a 2D reservoir which is connected with the EnKF method as a data assimilation technique. We will use different geostatistical properties for the Gaussian fields and different levels of the uncertainty introduced in the model parameters and also in the construction of the Gaussian fields.     

History matching on the Imperial College fault model using parallel tempering

The history-matching inverse problem from petroleum engineering is analysed using the Imperial College fault model. This fault model produces a challenging inverse problem and is designed to show some of the problems which can occur whilst performing history-matching calculations on complicated geologies. It is shown that there can be multiple distinct geologies which match the history data. Furthermore, it is shown that the maximum-a-posteriori estimate does not correspond to the true geology in some cases. Both of these statements are corroborated via numerical examples where the parameter spaces are ?, ?³, ?⁷ and ?¹³. In addition, it is shown that the number of matches which agree with the data increases with dimension for these examples. It is also shown that the different matches can result in different reservoir management decision which, if incorrectly taken, would incur substantial financial penalties. All of these analyses are performed in a systematic manner, where it is shown that the standard algorithms can give a misleading answer. The history-matching problem is written as a minimisation problem, and it is shown that knowledge of all of the local minima is required. This presents significant computational issues as the resulting objective function is highly nonlinear, expensive to evaluate and multimodal. Previously used algorithms have been proved to be inadequate. Parallel tempering is a method which, if run for long enough, can find all the local minima. However, as the objective is expensive, a number of algorithm modifications had to be used to ensure convergence within a reasonable time. This new information is outlined in the paper. The algorithm as implemented produced results and new insights into this problem which were not suspected before. The results produced by this algorithm for the multimodal history-matching problem are superior to all other results of which we are aware. However, a considered amount of computation time was used within this paper, so this result does not infer that the algorithm cannot be improved upon. This algorithm not only produces good results but can be applied to all other history-matching problems. We have shown that this method provides a robust route of finding multiple local optima/solutions to the inverse problem, which is of considerable benefit to the petroleum industry. Furthermore, it is an entirely parallel algorithm which is becoming computationally feasible for other history-matching problems.     

用于拟合生产动态数据的现有储层模型重构(英文)

计算机能力的提升和历史拟合方面的最新进展促进了对先前建立的储层模型的重新检验。为了节省工程师和CPU的时间,我们开发了4种独特的算法,来允许无需重新进行储层研究而重建现有模型。这些算法涉及的技术包括:优化、松弛、Wiener滤波或序贯重构。基本上,它们被用来确定一个随机函数和一系列随机数。给定一个随机函数,一族随机数将产生一个实现,这个实现和现有的储层模型十分接近。一旦随机数已知,现有的储层模型将被提交到一个历史拟合过程中,以此来改进数据拟合度或者考虑新收集到的数据。我们关注的是先前建立的相储层模型。虽然我们对模型模拟的方式一无所知,但是我们可以确定一系列随机数,再用多点统计模拟方法来建造一个和现有储层模型十分接近的实现。然后运行一种新的历史拟合程序来更新现有的储层模型,使其拟合两口新生产井的流量数据。     

Hybridization of the probability perturbation method with gradient information

Geostatistically based history-matching methods make it possible to devise history-matching strategies that will honor geologic knowledge about the reservoir. However, the performance of these methods is known to be impeded by slow convergence rates resulting from the stochastic nature of the algorithm. It is the purpose of this paper to introduce a method that integrates qualitative gradient information into the probability perturbation method to improve convergence. The potential of the proposed method is demonstrated on a synthetic history-matching example. The results indicate that inclusion of qualitative gradient information improves the performance of the probability perturbation method.     

Gaussian Processes for history-matching: application to an unconventional gas reservoir

The process of reservoir history-matching is a costly task. Many available history-matching algorithms either fail to perform such a task or they require a large number of simulation runs. To overcome such struggles, we apply the Gaussian Process (GP) modeling technique to approximate the costly objective functions and to expedite finding the global optima. A GP model is a proxy, which is employed to model the input-output relationships by assuming a multi-Gaussian distribution on the output values. An infill criterion is used in conjunction with a GP model to help sequentially add the samples with potentially lower outputs. The IC fault model is used to compare the efficiency of GP-based optimization method with other typical optimization methods for minimizing the objective function. In this paper, we present the applicability of using a GP modeling approach for reservoir history-matching problems, which is exemplified by numerical analysis of production data from a horizontal multi-stage fractured tight gas condensate well. The results for the case that is studied here show a quick convergence to the lowest objective values in less than 100 simulations for this 20-dimensional problem. This amounts to an almost 10 times faster performance compared to the Differential Evolution (DE) algorithm that is also known to be a powerful optimization technique. The sensitivities are conducted to explain the performance of the GP-based optimization technique with various correlation functions.     

Non-parametric Bayesian networks for parameter estimation in reservoir simulation: a graphical take on the ensemble Kalman filter (part I)

Reservoir simulation models are used both in the development of new fields and in developed fields where production forecasts are needed for investment decisions. When simulating a reservoir, one must account for the physical and chemical processes taking place in the subsurface. Rock and fluid properties are crucial when describing the flow in porous media. In this paper, the authors are concerned with estimating the permeability field of a reservoir. The problem of estimating model parameters such as permeability is often referred to as a history-matching problem in reservoir engineering. Currently, one of the most widely used methodologies which address the history-matching problem is the ensemble Kalman filter (EnKF). EnKF is a Monte Carlo implementation of the Bayesian update problem. Nevertheless, the EnKF methodology has certain limitations that encourage the search for an alternative method.For this reason, a new approach based on graphical models is proposed and studied. In particular, the graphical model chosen for this purpose is a dynamic non-parametric Bayesian network (NPBN). This is the first attempt to approach a history-matching problem in reservoir simulation using a NPBN-based method. A two-phase, two-dimensional flow model was implemented for a synthetic reservoir simulation exercise, and initial results are shown. The methods’ performances are evaluated and compared. This paper features a completely novel approach to history matching and constitutes only the first part (part I) of a more detailed investigation. For these reasons (novelty and incompleteness), many questions are left open and a number of recommendations are formulated, to be investigated in part II of the same paper.     

Theoretical connections between optimization algorithms based on an approximate gradient

Performing a line search method in the direction given by the simplex gradient is a well-known method in the mathematical optimization community. For reservoir engineering optimization problems, both a modification of the simultaneous perturbation stochastic approximation (SPSA) and ensemble-based optimization (EnOpt) have recently been applied for estimating optimal well controls in the production optimization step of closed-loop reservoir management. The modified SPSA algorithm has also been applied to assisted history-matching problems. A recent comparison of the performance of EnOpt and a SPSA-type algorithm (G-SPSA) for a set of production optimization test problems showed that the two algorithms resulted in similar estimates of the optimal net-present-value and required roughly the same amount of computational time to achieve these estimates. Here, we show that, theoretically, this result is not surprising. In fact, we show that both the simplex, preconditioned simplex, and EnOpt algorithms can be derived directly from a modified SPSA-type algorithm where the preconditioned simplex algorithm is presented for the first time in this paper. We also show that the expectation of all these preconditioned stochastic gradients is a first-order approximation of the preconditioning covariance matrix times the true gradient or a covariance matrix squared times the true gradient.     

A multiscale method for distributed parameter estimation with application to reservoir history matching

A method for multiscale parameter estimation with application to reservoir history matching is presented. Starting from a given fine-scale model, coarser models are generated using a global upscaling technique where the coarse models are tuned to match the solution of the fine model. Conditioning to dynamic data is done by history-matching the coarse model. Using consistently the same resolution both for the forward and inverse problems, this model is successively refined using a combination of downscaling and history matching until model-matching dynamic data are obtained at the finest scale. Large-scale corrections are obtained using fast models, which, combined with a downscaling procedure, provide a better initial model for the final adjustment on the fine scale. The result is thus a series of models with different resolution, all matching history as good as possible with this grid. Numerical examples show that this method may significantly reduce the computational effort and/or improve the quality of the solution when achieving a fine-scale match as compared to history-matching directly on the fine scale.     

Multiscale Parameterization of Petrophysical Properties for Efficient History-Matching

The prediction of fluid flows within hydrocarbon reservoirs requires the characterization of petrophysical properties. Such characterization is performed on the basis of geostatistics and history-matching; in short, a reservoir model is first randomly drawn, and then sequentially adjusted until it reproduces the available dynamic data. Two main concerns typical of the problem under consideration are the heterogeneity of rocks occurring at all scales and the use of data of distinct resolution levels. Therefore, referring to sequential Gaussian simulation, this paper proposes a new stochastic simulation method able to handle several scales for both continuous or discrete random fields. This method adds flexibility to history-matching as it boils down to the multiscale parameterization of reservoir models. In other words, reservoir models can be updated at either coarse or fine scales, or both. Parameterization adapts to the available data; the coarser the scale targeted, the smaller the number of unknown parameters, and the more efficient the history-matching process. This paper focuses on the use of variational optimization techniques driven by the gradual deformation method to vary reservoir models. Other data assimilation methods and perturbation processes could have been envisioned as well. Last, a numerical application case is presented in order to highlight the advantages of the proposed method for conditioning permeability models to dynamic data. For simplicity, we focus on two-scale processes. The coarse scale describes the variations in the trend while the fine scale characterizes local variations around the trend. The relationships between data resolution and parameterization are investigated.     

History matching of approximated lithofacies models under uncertainty

In history matching of lithofacies reservoir model, we attempt to find multiple realizations of lithofacies configuration that are conditional to dynamic data and representative of the model uncertainty space. This problem can be formalized in the Bayesian framework. Given a truncated Gaussian model as a prior and the dynamic data with its associated measurement error, we want to sample from the conditional distribution of the facies given the data. A relevant way to generate conditioned realizations is to use Markov chains Monte Carlo (MCMC). However, the dimensions of the model and the computational cost of each iteration are two important pitfalls for the use of MCMC. Furthermore, classical MCMC algorithms mix slowly, that is, they will not explore the whole support of the posterior in the time of the simulation. In this paper, we extend the methodology already described in a previous work to the problem of history matching of a Gaussian-related lithofacies reservoir model. We first show how to drastically reduce the dimension of the problem by using a truncated Karhunen-Loève expansion of the Gaussian random field underlying the lithofacies model. Moreover, we propose an innovative criterion of the choice of the number of components based on the connexity function. Then, we show how we improve the mixing properties of classical single MCMC, without increasing the global computational cost, by the use of parallel interacting Markov chains. Applying the dimension reduction and this innovative sampling method drastically lowers the number of iterations needed to sample efficiently from the posterior. We show the encouraging results obtained when applying the methodology to a synthetic history-matching case.     

Application of EM algorithms for seismic facices classification

Identification of the geological facies and their distribution from seismic and other available geological information is important during the early stage of reservoir development (e.g. decision on initial well locations). Traditionally, this is done by manually inspecting the signatures of the seismic attribute maps, which is very time-consuming. This paper proposes an application of the Expectation-Maximization (EM) algorithm to automatically identify geological facies from seismic data. While the properties within a certain geological facies are relatively homogeneous, the properties between geological facies can be rather different. Assuming that noisy seismic data of a geological facies, which reflect rock properties, can be approximated with a Gaussian distribution, the seismic data of a reservoir composed of several geological facies are samples from a Gaussian mixture model. The mean of each Gaussian model represents the average value of the seismic data within each facies while the variance gives the variation of the seismic data within a facies. The proportions in the Gaussian mixture model represent the relative volumes of different facies in the reservoir. In this setting, the facies classification problem becomes a problem of estimating the parameters defining the Gaussian mixture model. The EM algorithm has long been used to estimate Gaussian mixture model parameters. As the standard EM algorithm does not consider spatial relationship among data, it can generate spatially scattered seismic facies which is physically unrealistic. We improve the standard EM algorithm by adding a spatial constraint to enhance spatial continuity of the estimated geological facies. By applying the EM algorithms to acoustic impedance and Poisson’s ratio data for two synthetic examples, we are able to identify the facies distribution.     

Deterministic ensemble smoother with multiple data assimilation as an alternative for history-matching seismic data

This paper reports the results of an investigation on the use of a deterministic analysis scheme combined with the method ensemble smoother with multiple data assimilation (ES-MDA) for the problem of assimilating a large number of correlated data points. This is the typical case when history-matching time-lapse seismic data in petroleum reservoir models. The motivation for the use of the deterministic analysis is twofold. First, it tends to result in a smaller underestimation of the ensemble variance after data assimilation. This is particularly important for problems with a large number of measurements. Second, the deterministic analysis avoids the factorization of a large covariance matrix required in the standard implementation of ES-MDA with the perturbed observations scheme. The deterministic analysis is tested in a synthetic history-matching problem to assimilate production and seismic data.     

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.     

Upscaling facies models to preserve connectivity of designated facies

An upscaling algorithm has been developed that generates an irregular coarse grid that preserves flow connectivity by applying a rule-based upscaling algorithm to a fine-scale facies distribution. The algorithm is demonstrated using stochastically generated paleo-fluvial facies distributions. First, an irregular grid honoring the channel facies is created, followed by computation of effective anisotropic parameters for all coarse-grid cells. For the apparent layer-cake geometry of overbank deposits seen in outcrop, two local upscaling methods are compared: (1) the layered system approximation and (2) the mode. To assess upscaling performance, flow simulations for the original and upscaled grids are compared. The horizontal layered approximation (arithmetic mean) performs poorly, over-predicting lateral connectivity where even infrequent disconnection becomes important. Performance of the mode as an upscaling algorithm depends on the probability that a coarse-grid cell will be dominated by a single facies, and it performs surprisingly well because the upscaled grid-generation algorithm honors the channels, informing the upscaling process. Lastly, the irregular coarse grid was compared to a uniform coarse grid, showing superior performance with the irregular grid. The reduction in grid size achieved by irregular-grid generation will be a function of the geometrical complexity of the geologic objects to be honored.     

Stochastic simulation of categorical variables using a classification algorithm and simulated annealing

An important step of reservoir characterization is the stochastic modeling of the geometry of lithofacies which control large-scale heterogeneities of petrophysical properties. Although multiple realizations are necessary to appreciate the uncertainty in the spatial distribution of facies, a common short cut consists of retaining the first realization drawn. This paper presents an alternative to this potentially hazardous selection: (1) a categorical map is generated by allocating a single facies to each grid node according to the local probabilities of occurrence of the facies, and (2) the map then is post-processed using a steepest descent-type algorithm so as to improve reproduction of spatial continuity and transition probabilities between facies. The procedure is illustrated using a synthetic dataset. A waterflood simulation shows that retaining a single realization would yield, in average, larger errors in production forecasts (water cuts and recovered oil) than the single postprocessed facies map.     

Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.     

Geological Facies Recovery Based on Weighted $$\ell _1$$ ℓ 1 -Regularization

Mathematical Geosciences - A weighted compressed sensing (WCS) algorithm is proposed for the problem of channelized facies reconstruction from pixel-based measurements. This strategy integrates...     

利用元素地球化学特征研究河北平原第四纪沉积 环境的探讨与实践

应用河北省平原区多目标区域地球化学调查获得表层和深层土壤元素含量数据,对河北平原的第四纪沉积环境进行了分区。以SiO2、Na/Rb比值、Ca/Ba比值、Rb,U,Ga和Fe,Mg,Ni,V均一化累加和等地球化学指标作为第四纪沉积环境的分区指标,共划分出山前残坡积洪冲积相、冲积扇堆积相、中部平原河道带相、古黄河泛洪相、风积相、海陆过渡相等6个沉积相及河床亚相、漫滩亚相及洼地亚相等3个沉积亚相。此项研究是多目标地球化学应用于基础地学研究的重大实践,取得了十分理想的研究效果。     

Depositional and Diagenetic Controls on Sandstone Reservoirs with Low Porosity and Low Permeability in the Eastern Sulige Gas Field,China

In order to determine the genesis and the factors that control the low-porosity and low- permeability sandstone reservoirs in the eastern Sulige Gas Field in the Ordos Basin, systematic studies on the sedimentary facies and diagenesis were conducted by means of analysis of cores, thin sections, fluid inclusions, X-ray diffraction, cathode luminescence and scanning electron microscope. It was found that the sand bodies of the major gas reservoirs in the Shan1 section （P1S1） and the He8 section （P2H8） were formed during the Permian as sedimentary facies such as braided-channel bars, braided-river channels and point bars of a meandering river. Four types of diagenetic facies developed subsequently： in order from the best to the poorest properties these are type A （weak compaction, early calcite cement-chlorite film facies）, type B （moderate compaction, quartz overgrowth-feldspar corrosion-kaolinite filling facies）, type C （strong compaction, late calcite cement-quartz corrosion facies） and type D （matrix filling and strong compaction facies）. This diagenesis is undoubtedly the main reason for the poor reservoir properties of sandstone reservoirs, but the sedimentary facies are the underlying factors that greatly affect the diagenesis and thus the reservoir performance. Favorable diagenetic facies developed mainly in relatively small lithofacies such as braided-river channels, channel bars and point bars. The vertical distribution of the physical properties and the diagenetic facies of the reservoirs are related to the stratigraphic succession. Most of the sandstones between mudstones and thin beds of sandstone are unfavorable diagenetic facies. Analyses indicate that siliceous cementation can hardly be stopped by hydrocarbon filling. Authigenic chlorite could hardly protect the primary porosity. It not only occupies pore space, but also blocks pathways through sandstone reservoirs, so that it has significant influence on the permeability. Authigenic chlorite cannot be used as a marker for a     

Construction of a fluvial facies knowledge graph and its application in sedimentary facies identification

Lithofacies paleogeography is a data-intensive discipline that involves the interpretation and compilation of sedimentary facies. Traditional sedimentary facies analysis is a labor-intensive task with the added complexity of using unstructured knowledge and unstandardized terminology. Therefore, it is very difficult for beginners or non-geology scholars who lack a systematic knowledge and experience in sedimentary facies analysis. These hurdles could be partly alleviated by having a standardized, structured, and systematic knowledge base coupled with an efficient automatic machine-assisted sedimentary facies identification system. To this end, this study constructed a knowledge system for fluvial facies and carried out knowledge representation. Components include a domain knowledge graph for types of fluvial facies (meandering, braided and other fluvial depositional environments) and their characteristic features (bedforms, grain size distribution, etc.) with visualization, a method for query and retrieval on a graph database platform, a hierarchical knowledge tree-structure, a data-mining clustering algorithm for machine-analysis of publication texts, and an algorithm model for this area of sedimentary facies reasoning. The underlying sedimentary facies identification and knowledge reasoning system is based on expert experience and synthesis of publications. For testing, 17 sets literature publications data that included details of sedimentary facies data (bedforms, grain sizes, etc.) were submitted to the artificial intelligence model, then compared and validated. This testing set of automated reasoning results yielded an interpretation accuracy of about 90% relative to the published interpretations in those papers. Therefore, the model and algorithm provide an efficient and automated reasoning technology, which provides a new approach and route for the rapid and intelligent identification of other types of sedimentary facies from literature data or direct use in the field.