Evaluation of Gaussian approximations for data assimilation in reservoir models

The Bayesian framework is the standard approach for data assimilation in reservoir modeling. This framework involves characterizing the posterior distribution of geological parameters in terms of a given prior distribution and data from the reservoir dynamics, together with a forward model connecting the space of geological parameters to the data space. Since the posterior distribution quantifies the uncertainty in the geologic parameters of the reservoir, the characterization of the posterior is fundamental for the optimal management of reservoirs. Unfortunately, due to the large-scale highly nonlinear properties of standard reservoir models, characterizing the posterior is computationally prohibitive. Instead, more affordable ad hoc techniques, based on Gaussian approximations, are often used for characterizing the posterior distribution. Evaluating the performance of those Gaussian approximations is typically conducted by assessing their ability at reproducing the truth within the confidence interval provided by the ad hoc technique under consideration. This has the disadvantage of mixing up the approximation properties of the history matching algorithm employed with the information content of the particular observations used, making it hard to evaluate the effect of the ad hoc approximations alone. In this paper, we avoid this disadvantage by comparing the ad hoc techniques with a fully resolved state-of-the-art probing of the Bayesian posterior distribution. The ad hoc techniques whose performance we assess are based on (1) linearization around the maximum a posteriori estimate, (2) randomized maximum likelihood, and (3) ensemble Kalman filter-type methods. In order to fully resolve the posterior distribution, we implement a state-of-the art Markov chain Monte Carlo (MCMC) method that scales well with respect to the dimension of the parameter space, enabling us to study realistic forward models, in two space dimensions, at a high level of grid refinement. Our implementation of the MCMC method provides the gold standard against which the aforementioned Gaussian approximations are assessed. We present numerical synthetic experiments where we quantify the capability of each of the ad hoc Gaussian approximation in reproducing the mean and the variance of the posterior distribution (characterized via MCMC) associated to a data assimilation problem. Both single-phase and two-phase (oil–water) reservoir models are considered so that fundamental differences in the resulting forward operators are highlighted. The main objective of our controlled experiments was to exhibit the substantial discrepancies of the approximation properties of standard ad hoc Gaussian approximations. Numerical investigations of the type we present here will lead to the greater understanding of the cost-efficient, but ad hoc, Bayesian techniques used for data assimilation in petroleum reservoirs and hence ultimately to improved techniques with more accurate uncertainty quantification.     

基于IBUNE方法的水文模型不确定性分析

在贝叶斯理论框架下,根据一种可结合多个水文模型给出模拟或预报结果的IBUNE方法探讨了水文模型的输入、参数以及结构的不确定性问题。将SCEM-UA算法和EM算法嵌入新安江和TOPMODEL水文模型用于参数优化和模型平均,进而将输入与参数的综合不确定性处理后得到的预报量后验分布进行多模型综合,据此对水文模型的不确定性及其对水文模拟结果的影响进行评价。以湖南洣水流域龙家山水文站以上集水区域为例进行了应用研究,结果表明,IBUNE方法能够有效估计水文模型的不确定性,并能给出合理的概率预报区间。     

Anisotropic Continuous-in-Scale Universal Multifractal Cascades: Simulation,Analysis and Correction Methods

Continuous-in-scale multifractal cascades has long been an attractive choice for mathematically modeling turbulent and turbulent-like geophysical fields. These fields are usually anisotropic as they are subject to both stratification and rotation, thereby questioning the isotropy assumption often made to model them. The self-affine and generalized scale invariance approaches to scaling are used here to introduce anisotropy in such models. These anisotropic simulations have (1) unresolved large-scale features and (2) statistics that deviate from the desired power-law scaling mainly in the small scales. The former issue is solved via nesting, whereas the latter is attempted to be overcome using singularity correction methods. While earlier studies have proposed isotropic correction methods, here they have been generalized to correct anisotropic simulations. These singularity corrections seem to improve the small-scale statistical properties of mildly anisotropic simulations; nesting, on the other hand, appears to enhance statistics over almost all scales even for strongly anisotropic simulations. Both the correction and nesting techniques lead to a reduction in computational time and memory usage suggesting that nested singularity-corrected cascades offer a better framework for quantitatively modeling the atmosphere, ocean, solid earth, and associated fields.     

Prediction of porosity and water saturation using pre-stack seismic attributes: a comparison of Bayesian inversion and computational intelligence methods

Rock physical parameters such as porosity and water saturation play an important role in the mechanical behavior of hydrocarbon reservoir rocks. A valid and reliable prediction of these parameters from seismic data is essential for reservoir characterization, management, and also geomechanical modeling. In this paper, the application of conventional methods such as Bayesian inversion and computational intelligence methods, namely support vector regression (SVR) optimized by particle swarm optimization (PSO) and adaptive network-based fuzzy inference system-subtractive clustering method (ANFIS-SCM), is demonstrated to predict porosity and water saturation. The prediction abilities offered by Bayesian inversion, SVR-PSO, and ANFIS-SCM were presented using a synthetic dataset and field data available from a gas carbonate reservoir in Iran. In these models, seismic pre-stack data and attributes were utilized as the input parameters, while the porosity and water saturation were the output parameters. Various statistical performance indexes were utilized to compare the performance of those estimation models. The results achieved indicate that the ANFIS-SCM model has strong potential for indirect estimation of porosity and water saturation with high degree of accuracy and robustness from seismic data and attributes in both synthetic and real cases of this study.     

黑河出山径流的非线性特征分析

应用非线性动力学的理论和方法,对黑河出山径流的非线性特征进行了分析.结果表明,黑河月出山径流的年内分布、年平均流量的一次峰、谷变化符合单重或双重威布尔分布,并具有自相似性质.黑河出山径流多年变化在相空间中的运动轨迹收缩到一个约为4.32维的吸引子上,而描述流量的动力方程需要8个独立变量.黑河出山径流的非线性特征还表现在对内部结构为非线性函数的输入输出模型的良好应用上,如GRNN神经网络模型、非线性回归模型等.     

Conditioning generative adversarial networks on nonlinear data for subsurface flow model calibration and uncertainty quantification

Conditioning complex subsurface flow models on nonlinear data is complicated by the need to preserve the expected geological connectivity patterns to maintain solution plausibility. Generative adversarial networks (GANs) have recently been proposed as a promising approach for low-dimensional representation of complex high-dimensional images. The method has also been adopted for low-rank parameterization of complex geologic models to facilitate uncertainty quantification workflows. A difficulty in adopting these methods for subsurface flow modeling is the complexity associated with nonlinear flow data conditioning. While conditional GAN (CGAN) can condition simulated images on labels, application to subsurface problems requires efficient conditioning workflows for nonlinear data, which is far more complex. We present two approaches for generating flow-conditioned models with complex spatial patterns using GAN. The first method is through conditional GAN, whereby a production response label is used as an auxiliary input during the training stage of GAN. The production label is derived from clustering of the flow responses of the prior model realizations (i.e., training data). The underlying assumption of this approach is that GAN can learn the association between the spatial features corresponding to the production responses within each cluster. An alternative method is to use a subset of samples from the training data that are within a certain distance from the observed flow responses and use them as training data within GAN to generate new model realizations. In this case, GAN is not required to learn the nonlinear relation between production responses and spatial patterns. Instead, it is tasked to learn the patterns in the selected realizations that provide a close match to the observed data. The conditional low-dimensional parameterization for complex geologic models with diverse spatial features (i.e., when multiple geologic scenarios are plausible) performed by GAN allows for exploring the spatial variability in the conditional realizations, which can be critical for decision-making. We present and discuss the important properties of GAN for data conditioning using several examples with increasing complexity.

     

Mortar coupling and upscaling of pore-scale models

Pore-scale models are becoming increasingly useful as predictive tools for modeling flow and transport in porous media. These models can accurately represent the 3D pore-structure of real media. Currently first-principles modeling methods are being employed for obtaining qualitative and quantitative behavior. Generally, artificial, simple boundary conditions are imposed on a model that is used as a stand-alone tool for extracting macroscopic parameters. However, realistic boundary conditions, reflecting flow and transport in surrounding media, may be necessary for behavior that occurs over larger length scales or including pore-scale models in a multiscale setting. Here, pore-scale network models are coupled to adjacent media (additional pore-scale or continuum-scale models) using mortars. Mortars are 2D finite-element spaces employed to couple independent subdomains by enforcing continuity of pressure and flux at shared boundary interfaces. While mortars have been used in the past to couple subdomains of different models, physics, and meshes, they are extended here for the first time to pore-scale models. The approach is demonstrated by modeling single-phase flow in coupled pore-scale models, but the methodology can be utilized to model dynamic processes and perform multiscale modeling in 3D continuum simulators for flow and transport.     

基于田间试验的土壤水分运动模型选择

基于不同形式Richards方程可建立不同适用范围和计算精度的数值模型,针对具体情况下如何选择合适模型的问题,以武汉大学农田水利试验场田间入渗试验为例,选用6种模型(Picard-h模型、Picard-θ模型、Picard-mix模型、Ross模型、动力波模型和水均衡模型),运用贝叶斯模型平均(BMA)方法进行了模型选择的计算;针对BMA方法无法考虑模型计算效率的缺点,进一步提出了可同时考虑模型计算精度与计算效率的改进BMA方法。计算结果表明,在本田间尺度问题中,Ross模型排序最高,说明其兼具高精度与高效率,改进BMA方法可增加高计算效率模型被选中的概率,使模型选择更加全面合理。     

Probabilistic Methods Applied to Unsaturated Numerical Modeling

The numerical modeling of unsaturated soil processes is becoming more prevalent worldwide. Although numerical modeling is becoming increasingly accepted in geotechnical engineering practice, care must be exercised and improper modeling techniques and procedures must be avoided. Many issues such as nodal resolution and imperfect convergence can result in inaccurate solutions. Unfortunately, analyses of highly nonlinear unsaturated soil flow and slope stability models can significantly increase the modeling time required. As a result, there is a trend to reduce the number of model runs. Results are often presented to client as single model runs or simplistic sensitivity analysis. This paper presents methodologies for applying probabilistic methods to unsaturated soils seepage and slope stability analysis models. The focus is on the application of the alternative point estimate method to practical problems in such a way as to minimize the number of model runs. The demonstration of a successful application to a waste rock pile is presented.     

Bayesian updating via bootstrap filtering combined with data-driven polynomial chaos expansions: methodology and application to history matching for carbon dioxide storage in geological formations

Model calibration and history matching are important techniques to adapt simulation tools to real-world systems. When prediction uncertainty needs to be quantified, one has to use the respective statistical counterparts, e.g., Bayesian updating of model parameters and data assimilation. For complex and large-scale systems, however, even single forward deterministic simulations may require parallel high-performance computing. This often makes accurate brute-force and nonlinear statistical approaches infeasible. We propose an advanced framework for parameter inference or history matching based on the arbitrary polynomial chaos expansion (aPC) and strict Bayesian principles. Our framework consists of two main steps. In step 1, the original model is projected onto a mathematically optimal response surface via the aPC technique. The resulting response surface can be viewed as a reduced (surrogate) model. It captures the model’s dependence on all parameters relevant for history matching at high-order accuracy. Step 2 consists of matching the reduced model from step 1 to observation data via bootstrap filtering. Bootstrap filtering is a fully nonlinear and Bayesian statistical approach to the inverse problem in history matching. It allows to quantify post-calibration parameter and prediction uncertainty and is more accurate than ensemble Kalman filtering or linearized methods. Through this combination, we obtain a statistical method for history matching that is accurate, yet has a computational speed that is more than sufficient to be developed towards real-time application. We motivate and demonstrate our method on the problem of CO₂ storage in geological formations, using a low-parametric homogeneous 3D benchmark problem. In a synthetic case study, we update the parameters of a CO₂/brine multiphase model on monitored pressure data during CO₂ injection.     

Ensemble clustering for efficient robust optimization of naturally fractured reservoirs

In the development of naturally fractured reservoirs (NFRs), the existence of natural fractures induces severe fingering and breakthrough. To manage the flooding process and improve the ultimate recovery, we propose a numerical workflow to generate optimal production schedules for smart wells, in which the inflow control valve (ICV) settings can be controlled individually. To properly consider the uncertainty introduced by randomly distributed natural fractures, the robust optimization would require a large ensemble size and it would be computationally demanding. In this work, a hierarchical clustering method is proposed to select representative models for the robust optimization in order to avoid redundant simulation runs and improve the efficiency of the robust optimization. By reducing the full ensemble of models into a small subset ensemble, the efficiency of the robust optimization algorithm is significantly improved. The robust optimization is performed using the StoSAG scheme to find the optimal well controls that maximize the net-present-value (NPV) of the NFR’s development. Due to the discrete property of a natural fracture field, traditional feature extraction methods such as model-parameter-based clustering may not be directly applicable. Therefore, two different kinds of clustering-based optimization methods, a state-based (e.g., s _w profiles) clustering and a response-based (e.g., production rates) clustering, are proposed and compared. The computational results show that the robust clustering optimization could increase the computational efficiency significantly without sacrificing much expected NPV of the robust optimization. Moreover, the performance of different clustering algorithms varies widely in correspondence to different selections of clustering features. By properly extracting model features, the clustered subset could adequately represent the uncertainty of the full ensemble.     

Bayesian analysis of data and model uncertainty in 3D seismic travel-time tomography

We present a probabilistic analysis of seismic travel-time equations using the Bayesian Method. The assessment of models and data is crucial in 3D seismic travel-time tomography, and a method quantitatively assess the quality of both the data and the model is necessary in order to attain the most realistic results. The Bayesian method that we propose here is more effective than the frequentist approach, both in analysis time and uncertainty minimization, when processing large sets of tomographic data.     

Versatile Two-Level Schwarz Preconditioners for Multiphase Flow

Numerically modeling groundwater flow on finely discretized two- and three-dimensional domains requires solution algorithms appropriate for distributed memory multiprocessor architectures. Multilevel and domain decomposition algorithms are appropriate for preconditioning or solving linear systems in parallel and have, therefore, been applied to linear models for saturated groundwater flow. These algorithms have also been incorporated into more complex nonlinear multiphase flow models in the context of a linearization procedure such as Newton's method. In this work, we study a class of parallel preconditioners based on two-level Schwarz domain decomposition applied in a nonlinear two-phase flow numerical model. The restriction and interpolation operators are based on an aggregation approach that has a straightforward implementation for a variety of applications arising in subsurface modeling: structured and unstructured discretizations, finite elements and finite differences, and multicomponent model equations. We present model formulations, results from numerical experiments, and a comparison of a standard one-level Schwarz method to three two-level aggregation-based methods.     

Accuracy and efficiency of time integration methods for 1D diffusive wave equation

The diffusive wave approximation of the Saint-Venant equations is commonly used in hydrological models to describe surface flow processes. Numerous numerical approaches can be used to solve this highly nonlinear equation. Nonlinear time integration schemes—also called methods of lines (MOL)—were proven very efficient to solve other nonlinear problems in geosciences but were never considered to deal with surface flow modeling with the diffusive wave equation. In this paper, we study the relative performance of different time and space integration schemes by comparing the results obtained with classical approaches and with nonlinear time integration approaches. The results show that (i) the integration method with a higher order in space shows high accuracy regarding an integrated indicator such as the global mass balance error but is less accurate regarding local indicators, and (ii) nonlinear time integration techniques perform better than classical ones. Overall, it seems that integration techniques combining nonlinear time integration and a low spatial order need to be considered when developing hydrological modeling tools owing to their simplicity of implementation and very good performance.     

电法测井的三维有限元模拟

研究了不同测量方式(井-地,地-井,井-井)下点源场井中电法的三维有限元数值模拟。考虑到深度方向上大范围的网格剖分和井眼的影响及井-井测量等因素,采用放射状三棱柱单元的网格剖分方式,以提高网格质量,减少剖分单元数;给出了三棱柱单元的坐标变换公式,进行精确的单元积分,减少了单元积分时间;结合非结构化网格技术,实现了复杂模型的模拟;开发出相应的程序实现了复杂条件下(如考虑井眼影响、井井测量、倾斜井情形、地形起伏等)电法测井的三维有限元模拟,数值算例验证了方法的可靠性及计算效率,并对不同情形下的异常响应进行了分析,为进一步的反演工作奠定了基础。     

顺序数据同化的Bayes滤波框架

数据同化是在动力学模型的运行过程中不断融合新的观测信息的方法论,Bayes理论是数据同化的基石.从原理、方法和符号系统为Bayes滤波在数据同化中的应用勾勒一个统一的框架.首先对连续数据同化和顺序数据同化的各种方法做了分类,然后给出了非线性系统顺序数据同化的Bayes递推滤波形式,并在此基础上介绍了典型的顺序数据同化方法--粒子滤波和集合Kalman滤波.粒子滤波实质上是一种基于递推Bayes估计和Monte Carlo模拟的滤波方法,而集合Kalman滤波相当于一种权值相等的粒子滤波.Bayes滤波理论为顺序数据同化提供了更广义的理论框架,从基础的数学理论上揭示了数据同化的基本原理.     

Time homogenization for clays subjected to large numbers of cycles

This paper discusses the reliability and the efficiency of a time homogenization method employed to reduce the computational time during cyclic loading for two common geotechnical tests and two elastoplastic models for clays. The method of homogenization is based upon splitting time into two separate scales. The first time scale relates to the period of cyclic loading and the second to the characteristic time of the fatigue phenomenon. The time homogenization method is applied to simulate an undrained triaxial test (homogeneous stress state) and a pressuremeter test (nonhomogeneous stress state) under one‐way cyclic loading on normally consolidated clay. This method is coupled with two elastoplastic models dedicated to cyclic behavior of clay (a bounding surface plasticity model and a bubble model). Both linear and nonlinear elasticities are considered. The difficulty encountered when applying this method to models introducing nonlinear elasticity and kinematic hardening is pointed out. The performance of time homogenization related to the main parameters is numerically investigated by comparison with conventional finite element simulations. Copyright © 2012 John Wiley & Sons, Ltd.     

基于核主成分支持向量机的火成岩QAPF分类——以青海格尔木地区为例

利用核主成分(KPCA)较强的非线性特征提取能力对Hyperion高光谱数据进行降维及光谱特征提取,将特征信息作为支持向量机(SVM)建模样本的观测数据,建立KPCA-SVM回归模型,利用该模型进行研究区岩石氧化物百分含量反演。同时,依据国际地质科学联合会提出的QAPF火成岩分类方案对区内火成岩进行了岩性划分。研究结果表明:KPCA降维后的高光谱数据反演氧化物含量的效果良好;而基于QAPF模型的火成岩划分结果也十分理想,分类结果对已有地质图进行了有效的补充。KPCA-SVM理论模型为利用高光谱遥感数据进行岩性分类提供了一种快速可行的方法。