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.     

Multimodal ensemble Kalman filtering using Gaussian mixture models

In this paper we present an extension of the ensemble Kalman filter (EnKF) specifically designed for multimodal systems. EnKF data assimilation scheme is less accurate when it is used to approximate systems with multimodal distribution such as reservoir facies models. The algorithm is based on the assumption that both prior and posterior distribution can be approximated by Gaussian mixture and it is validated by the introduction of the concept of finite ensemble representation. The effectiveness of the approach is shown with two applications. The first example is based on Lorenz model. In the second example, the proposed methodology combined with a localization technique is used to update a 2D reservoir facies models. Both applications give evidence of an improved performance of the proposed method respect to the EnKF.     

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.     

History matching of statistically anisotropic fields using the Karhunen-Loeve expansion-based global parameterization technique

Traditional ensemble-based history matching method, such as the ensemble Kalman filter and iterative ensemble filters, usually update reservoir parameter fields using numerical grid-based parameterization. Although a parameter constraint term in the objective function for deriving these methods exists, it is difficult to preserve the geological continuity of the parameter field in the updating process of these methods; this is especially the case in the estimation of statistically anisotropic fields (such as a statistically anisotropic Gaussian field and facies field with elongated facies) with uncertainties about the anisotropy direction. In this work, we propose a Karhunen-Loeve expansion-based global parameterization technique that is combined with the ensemble-based history matching method for inverse modeling of statistically anisotropic fields. By using the Karhunen-Loeve expansion, a Gaussian random field can be parameterized by a group of independent Gaussian random variables. For a facies field, we combine the Karhunen-Loeve expansion and the level set technique to perform the parameterization; that is, for each facies, we use a Gaussian random field and a level set algorithm to parameterize it, and the Gaussian random field is further parameterized by the Karhunen-Loeve expansion. We treat the independent Gaussian random variables in the Karhunen-Loeve expansion as the model parameters. When the anisotropy direction of the statistically anisotropic field is uncertain, we also treat it as a model parameter for updating. After model parameterization, we use the ensemble randomized maximum likelihood filter to perform history matching. Because of the nature of the Karhunen-Loeve expansion, the geostatistical characteristics of the parameter field can be preserved in the updating process. Synthetic cases are set up to test the performance of the proposed method. Numerical results show that the proposed method is suitable for estimating statistically anisotropic fields.     

A multi-resolution workflow to generate high-resolution models constrained to dynamic data

Distance-based stochastic techniques have recently emerged in the context of ensemble modeling, in particular for history matching, model selection and uncertainty quantification. Starting with an initial ensemble of realizations, a distance between any two models is defined. This distance is defined such that the objective of the study is incorporated into the geological modeling process, thereby potentially enhancing the efficacy of the overall workflow. If the intent is to create new models that are constrained to dynamic data (history matching), the calculation of the distance requires flow simulation for each model in the initial ensemble. This can be very time consuming, especially for high-resolution models. In this paper, we present a multi-resolution framework for ensemble modeling. A distance-based procedure is employed, with emphasis on the rapid construction of multiple models that have improved dynamic data conditioning. Our intent is to construct new high-resolution models constrained to dynamic data, while performing most of the flow simulations only on upscaled models. An error modeling procedure is introduced into the distance calculations to account for potential errors in the upscaling. Based on a few fine-scale flow simulations, the upscaling error is estimated for each model using a clustering technique. We demonstrate the efficiency of the method on two examples, one where the upscaling error is small, and another where the upscaling error is significant. Results show that the error modeling procedure can accurately capture the error in upscaling, and can thus reproduce the fine-scale flow behavior from coarse-scale simulations with sufficient accuracy (in terms of uncertainty predictions). As a consequence, an ensemble of high-resolution models, which are constrained to dynamic data, can be obtained, but with a minimum of flow simulations at the fine scale.     

Application of the EnKF and Localization to Automatic History Matching of Facies Distribution and Production Data

The performance of the ensemble Kalman filter (EnKF) for continuous updating of facies location and boundaries in a reservoir model based on production and facies data for a 3D synthetic problem is presented. The occurrence of the different facies types is treated as a random process and the initial distribution was obtained by truncating a bi-Gaussian random field. Because facies data are highly non-Gaussian, re-parameterization was necessary in order to use the EnKF algorithm for data assimilation; two Gaussian random fields are updated in lieu of the static facies parameters. The problem of history matching applied to facies is difficult due to (1) constraints to facies observations at wells are occasionally violated when productions data are assimilated; (2) excessive reduction of variance seems to be a bigger problem with facies than with Gaussian random permeability and porosity fields; and (3) the relationship between facies variables and data is so highly non-linear that the final facies field does not always honor early production data well. Consequently three issues are investigated in this work. Is it possible to iteratively enforce facies constraints when updates due to production data have caused them to be violated? Can localization of adjustments be used for facies to prevent collapse of the variance during the data-assimilation period? Is a forecast from the final state better than a forecast from time zero using the final parameter fields?To investigate these issues, a 3D reservoir simulation model is coupled with the EnKF technique for data assimilation. One approach to enforcing the facies constraint is continuous iteration on all available data, which may lead to inconsistent model states, incorrect weighting of the production data and incorrect adjustment of the state vector. A sequential EnKF where the dynamic and static data are assimilated sequentially is presented and this approach seems to have solved the highlighted problems above. When the ensemble size is small compared to the number of independent data, the localized adjustment of the state vector is a very important technique that may be used to mitigate loss of rank in the ensemble. Implementing a distance-based localization of the facies adjustment appears to mitigate the problem of variance deficiency in the ensembles by ensuring that sufficient variability in the ensemble is maintained throughout the data assimilation period. Finally, when data are assimilated without localization, the prediction results appear to be independent of the starting point. When localization is applied, it is better to predict from the start using the final parameter field rather than continue from the final state.     

多点统计地质建模技术研究进展与应用*

多点统计地质建模技术自提出至今已有20余年的历史,已经成为储集层地质建模的国际前沿研究方向,在理论和应用研究方面都取得了长足进展。以多点统计地质建模技术的发展历程为主线,以多点统计地质建模技术的技术进展为核心,论述了多点统计地质建模技术的研究进展,对主要的多点统计地质建模方法进行了分类,系统讨论了具有发展潜力的多点统计地质建模方法的原理、特点以及存在的问题,并以扎格罗斯盆地孔隙型碳酸盐岩油藏S油藏为例进行了应用研究,对比了多点统计模拟与序贯指示模拟的优劣。研究表明,多点统计模拟在复杂的相模拟方面,较序贯指示模拟具有明显的优势;基于图型的Dispat方法采用图型替换数据事件的策略,使相的分布规律更符合地质学家的地质认识。这一认识为孔隙型碳酸盐岩油藏建模提供了一种新思路,对类似油藏的地质建模具有借鉴作用。     

Continuous Facies Updating Using the Ensemble Kalman Filter and the Level Set Method

We present a methodology based on the ensemble Kalman filter (EnKF) and the level set method for the continuous model updating of geological facies with respect to production data. Geological facies are modeled using an implicit surface representation and conditioned to production data using the ensemble Kalman filter. The methodology is based on Gaussian random fields used to deform the facies boundaries. The Gaussian random fields are used as the model parameter vector to be updated sequentially within the EnKF when new measurements are available. We show the successful application of the methodology to two synthetic reservoir models.     

Sedimentary Microfacies and Porosity Modeling of Deep-Water Sandy Debris Flows by Combining Sedimentary Patterns with Seismic Data: An Example from Unit I of Gas Field A, South China Sea

Sandy debris flow deposits are present in Unit I during Miocene of Gas Field A in the Baiyun Depression of the South China Sea. The paucity of well data and the great variability of the sedimentary microfacies make it difficult to identify and predict the distribution patterns of the main gas reservoir, and have seriously hindered further exploration and development of the gas field. Therefore, making full use of the available seismic data is extremely important for predicting the spatial distribution of sedimentary microfacies when constructing three-dimensional reservoir models. A suitable reservoir modeling strategy or workflow controlled by sedimentary microfacies and seismic data has been developed. Five types of seismic attributes were selected to correlate with the sand percentage, and the root mean square (RMS) amplitude performed the best. The relation between the RMS amplitude and the sand percentage was used to construct a reservoir sand distribution map. Three types of main sedimentary microfacies were identified: debris channels, fan lobes, and natural levees. Using constraints from the sedimentary microfacies boundaries, a sedimentary microfacies model was constructed using the sequential indicator and assigned value simulation methods. Finally, reservoir models of physical properties for sandy debris flow deposits controlled by sedimentary microfacies and seismic inversion data were established. Property cutoff values were adopted because the sedimentary microfacies and the reservoir properties from well-logging interpretation are intrinsically different. Selection of appropriate reservoir property cutoffs is a key step in reservoir modeling when using simulation methods based on sedimentary microfacies control. When the abnormal data are truncated and the reservoir properties probability distribution fits a normal distribution, microfacies-controlled reservoir property models are more reliable than those obtained from the sequence Gauss simulation method. The cutoffs for effective porosity of the debris channel, fan lobe, and natural levee facies were 0.2, 0.09, and 0.12, respectively; the corresponding average effective porosities were 0.24, 0.13, and 0.15. The proposed modeling method makes full use of seismic attributes and seismic inversion data, and also makes the property data of single-well depositional microfacies more conformable to a normal distribution with geological significance. Thus, the method allows use of more reliable input data when we construct a model of a sandy debris flow.     

Relation Between Level Set and Truncated Pluri-Gaussian Methodologies for Facies Representation

Truncated Gaussian/pluri-Gaussian representations and level set representations are two methodologies for implicit representation that can be applied to large-scale subsurface structures. Identification of facies in a petroleum reservoir from dynamic well data is a common field of application for these representation methodologies. No comparison of the methodologies has appeared in the literature to date. The paper seeks to improve on this situation by comparing selected level set and truncated Gaussian/pluri-Gaussian representations in detail. Strong similarities are found, in the sense that every truncated Gaussian/pluri-Gaussian representation considered has a level set counterpart. Furthermore, the transition from a truncated Gaussian/pluri-Gaussian representation to the corresponding level set representation is easily accessible. In addition to the comparison aspect, this paper also introduces a novel level set representation—the hierarchical level set representation—that removes a difficulty present in existing level set representations in a shape estimation setting. It is shown that the hierarchical level set representation corresponds to a well-known truncated pluri-Gaussian representation.     

河流相油气储层的井震结合相控随机建模约束方法

河流相油气储层的研究传统上多是只依据井点资料,先在井上进行沉积相的划分,而后进行剖面相的分析,最后再结合平面沉积参数等值线图编制平面相图,这样往往会造成"见砂画河,吾跟勘探走"的局面,这种平面相图在井间可能存在着较大的误差。然而,平面相图的正确与否直接影响着储层建模中相控的结果。为此作者提出了一种井震结合进行沉积相图编制的新方法,即"以河找砂,指导勘探行"的思路,并在此基础上进行分层次地相控约束随机建模。同时提出相控建模的三个基本的约束条件,即首先要保证随机建模模型的"相序"符合地质规律;其次要保证建模实现的微相分布统计概率与单井沉积微相数据离散化至三维网格后的统计概率相一致;第三要确保三维数据中每种微相的变差函数特征与定量地质知识库一致。因而,从沉积形成与演化的成因角度来指导沉积储层随机建模过程,应用多参数协同、分层次约束的方法,以河道的平面展布和垂向演化来控制建模的结果,使其更逼近地下地质的真实。     

Control Function Measures for Hydrodynamic Problems

This short paper shows that different choices made for control functions of information in hydrodynamic flow problems can have significant implications for interpretations of the system. Using as a simple illustration the case of a steady-state, one-dimensional flow model with no internal sources or sinks and with the hydraulic conductivity depending on a single parameter and the distance from the origin, it is shown that, even when a continuous, error-free head data set is provided, statements about the uniqueness or not of the inverse solution are conditioned on the choice of the control function. Care has to be exercised in obtaining physically meaningful results and, depending on the model assumptions and the data available, there may not be acceptable models. It is also shown that there may be more than one model behavior that is acceptable. The results have implications for the hydrodynamic upscaling problem for flow in permeable media, for ensemble averaging methods, and for parameter determination for deterministic models of permeable flow.     

Incorporating Information About Edge Effects When Simulating Lithofacies

Some years ago, earth scientists came to realize that knowing more about the geology of an orebody or an oil reservoir makes it easier to make the appropriate decisions concerning mine planning or reservoir exploitation. Geostatistical techniques for simulating lithofacies—that is, the geometry of the geology—were developed as a result of this. These methods should be able to produce geological images that respect not only the anisotropies of the different lithofacies but also their spatial layout relative to one another. While indicator variograms ensure that anisotropies are respected, another tool needs to be incorporated in the simulation technique to reflect the relative spatial layout of the different lithofacies. We propose to use the concept of edge effects that define the position of one lithofacies relative to another. Simple tests using direct and cross indicator variograms confirm the presence or absence of edge effects. We investigate if and how edge effect information can be incorporated in the different indicator simulation techniques—sequential indicator simulations, simulated annealing, the truncated Gaussian method and plurigaussian simulations. Results show that the choice of simulation method must be guided by the edge effect characteristics of the experimental lithologic data.     

GANSim: Conditional Facies Simulation Using an Improved Progressive Growing of Generative Adversarial Networks (GANs)

Conditional facies modeling combines geological spatial patterns with different types of observed data, to build earth models for predictions of subsurface resources. Recently, researchers have used generative adversarial networks (GANs) for conditional facies modeling, where an unconditional GAN is first trained to learn the geological patterns using the original GAN’s loss function, then appropriate latent vectors are searched to generate facies models that are consistent with the observed conditioning data. A problem with this approach is that the time-consuming search process needs to be conducted for every new conditioning data. As an alternative, we improve GANs for conditional facies simulation (called GANSim) by introducing an extra condition-based loss function and adjusting the architecture of the generator to take the conditioning data as inputs, based on progressive growing of GANs. The condition-based loss function is defined as the inconsistency between the input conditioning value and the corresponding characteristics exhibited by the output facies model, and forces the generator to learn the ability of being consistent with the input conditioning data, together with the learning of geological patterns. Our input conditioning factors include global features (e.g., the mud facies proportion) alone, local features such as sparse well facies data alone, and joint combination of global features and well facies data. After training, we evaluate both the quality of generated facies models and the conditioning ability of the generators, by manual inspection and quantitative assessment. The trained generators are quite robust in generating high-quality facies models conditioned to various types of input conditioning information.

     

Ensemble-Based Seismic and Production Data Assimilation Using Selection Kalman Model

Data assimilation in reservoir modeling often involves model variables that are multimodal, such as porosity and permeability. Well established data assimilation methods such as ensemble Kalman filter and ensemble smoother approaches, are based on Gaussian assumptions that are not applicable to multimodal random variables. The selection ensemble smoother is introduced as an alternative to traditional ensemble methods. In the proposed method, the prior distribution of the model variables, for example the porosity field, is a selection-Gaussian distribution, which allows modeling of the multimodal behavior of the posterior ensemble. The proposed approach is applied for validation on a two-dimensional synthetic channelized reservoir. In the application, an unknown reservoir model of porosity and permeability is estimated from the measured data. Seismic and production data are assumed to be repeatedly measured in time and the reservoir model is updated every time new data are assimilated. The example shows that the selection ensemble Kalman model improves the characterisation of the bimodality of the model parameters compared to the results of the ensemble smoother.

     

Ensemble filter methods with perturbed observations applied to nonlinear problems

The ensemble Kalman filter (EnKF) has been shown repeatedly to be an effective method for data assimilation in large-scale problems, including those in petroleum engineering. Data assimilation for multiphase flow in porous media is particularly difficult, however, because the relationships between model variables (e.g., permeability and porosity) and observations (e.g., water cut and gas–oil ratio) are highly nonlinear. Because of the linear approximation in the update step and the use of a limited number of realizations in an ensemble, the EnKF has a tendency to systematically underestimate the variance of the model variables. Various approaches have been suggested to reduce the magnitude of this problem, including the application of ensemble filter methods that do not require perturbations to the observed data. On the other hand, iterative least-squares data assimilation methods with perturbations of the observations have been shown to be fairly robust to nonlinearity in the data relationship. In this paper, we present EnKF with perturbed observations as a square root filter in an enlarged state space. By imposing second-order-exact sampling of the observation errors and independence constraints to eliminate the cross-covariance with predicted observation perturbations, we show that it is possible in linear problems to obtain results from EnKF with observation perturbations that are equivalent to ensemble square-root filter results. Results from a standard EnKF, EnKF with second-order-exact sampling of measurement errors that satisfy independence constraints (EnKF (SIC)), and an ensemble square-root filter (ETKF) are compared on various test problems with varying degrees of nonlinearity and dimensions. The first test problem is a simple one-variable quadratic model in which the nonlinearity of the observation operator is varied over a wide range by adjusting the magnitude of the coefficient of the quadratic term. The second problem has increased observation and model dimensions to test the EnKF (SIC) algorithm. The third test problem is a two-dimensional, two-phase reservoir flow problem in which permeability and porosity of every grid cell (5,000 model parameters) are unknown. The EnKF (SIC) and the mean-preserving ETKF (SRF) give similar results when applied to linear problems, and both are better than the standard EnKF. Although the ensemble methods are expected to handle the forecast step well in nonlinear problems, the estimates of the mean and the variance from the analysis step for all variants of ensemble filters are also surprisingly good, with little difference between ensemble methods when applied to nonlinear problems.