Machine-learning-based modeling of coarse-scale error,with application to uncertainty quantification

The use of upscaled models is attractive in many-query applications that require a large number of simulation runs, such as uncertainty quantification and optimization. Highly coarsened models often display error in output quantities of interest, e.g., phase production and injection rates, so the direct use of these results for quantitative evaluations and decision making may not be appropriate. In this work, we introduce a machine-learning-based post-processing framework for modeling the error in coarse-model results in the context of uncertainty quantification. Coarse-scale models are constructed using an accurate global single-phase transmissibility upscaling procedure. The framework entails the use of high-dimensional regression (random forest in this work) to model error based on a number of error indicators or features. Many of these features are derived from approximations of the subgrid effects neglected in the coarse-scale saturation equation. These features are identified through volume averaging, and they are generated by solving a fine-scale saturation equation with a constant-in-time velocity field. Our approach eliminates the need for the user to hand-design a small number of informative (relevant) features. The training step requires the simulation of some number of fine and coarse models (in this work we perform either 10 or 30 training simulations), followed by construction of a regression model for each well. Classification is also applied for production wells. The methodology then provides a correction at each time step, and for each well, in the phase production and injection rates. Results are presented for two- and three-dimensional oil–water systems. The corrected coarse-scale solutions show significantly better accuracy than the uncorrected solutions, both in terms of realization-by-realization predictions for oil and water production rates, and for statistical quantities important for uncertainty quantification, such as P10, P50, and P90 predictions.     

Permeability upscaling of fault zones in the Aztec Sandstone,Valley of Fire State Park,Nevada, with a focus on slip surfaces and slip bands

The effect of open and filled slip surfaces on the upscaled permeability of two fault zones with 6 and 14 m strike-slip in an eolian Aztec Sandstone, Nevada, USA is evaluated. Each fault zone is composed of several fault components: a fault core, bounded by filled through-going slip surfaces referred to as slip bands, and a surrounding damage zone that contains joints and deformation bands. Slip band geometry, composition, and petrophysical properties are characterized. Measurements and modeling show that slip band permeabilities can vary over 12 orders of magnitude depending on the degree of fill within the slip bands. The slip bands along with other fault zone components are represented in finite volume numerical calculations and the impact of various slip-band representations on upscaled fault zone permeability is tested. The results show 2 orders of magnitude variation in upscaled fault zone permeability in the fault-normal direction and a factor of 2 variation in the fault-parallel direction. The numerical results presented here are compared to the earlier numerical results in which structured Cartesian grids were used for the numerical simulations, and are in qualitative agreement with earlier calculations but use about a factor of 250–400 fewer numerical cells.     

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.     

A New Differentiable Parameterization Based on Principal Component Analysis for the Low-Dimensional Representation of Complex Geological Models

A new approach based on principal component analysis (PCA) for the representation of complex geological models in terms of a small number of parameters is presented. The basis matrix required by the method is constructed from a set of prior geological realizations generated using a geostatistical algorithm. Unlike standard PCA-based methods, in which the high-dimensional model is constructed from a (small) set of parameters by simply performing a multiplication using the basis matrix, in this method the mapping is formulated as an optimization problem. This enables the inclusion of bound constraints and regularization, which are shown to be useful for capturing highly connected geological features and binary/bimodal (rather than Gaussian) property distributions. The approach, referred to as optimization-based PCA (O-PCA), is applied here mainly for binary-facies systems, in which case the requisite optimization problem is separable and convex. The analytical solution of the optimization problem, as well as the derivative of the model with respect to the parameters, is obtained analytically. It is shown that the O-PCA mapping can also be viewed as a post-processing of the standard PCA model. The O-PCA procedure is applied both to generate new (random) realizations and for gradient-based history matching. For the latter, two- and three-dimensional systems, involving channelized and deltaic-fan geological models, are considered. The O-PCA method is shown to perform very well for these history matching problems, and to provide models that capture the key sand–sand and sand–shale connectivities evident in the true model. Finally, the approach is extended to generate bimodal systems in which the properties of both facies are characterized by Gaussian distributions. MATLAB code with the O-PCA implementation, and examples demonstrating its use are provided online as Supplementary Materials.     

Reflection of Oblique Incident Waves by Breakwaters with Partially-Perforated Wall

The reflection of oblique incident waves from breakwaters with a partially-perforated front wall is investigated. The fluid domain is divided into two sub-domains and the eigenfunction expansion method is applied to expand velocity poten-tials in each domain. In the eigen-expansion of the velocity potential, evanescent waves are included. Numerical results of the present model are compared with experimental data. The effect of porosity, the relative chamber width, the relative water depth in the wave absorbing chamber and the water depth in front of the structure are discussed.     

与发射线区成协的类星体CIV吸收系统的研究(Ⅰ)样本

高红移活动星系核的环境是目前亟待研究的领域,与类星体发射线区成协的吸收线系统有可能被使用.为此我们对CIV成协吸收系统进行了统计研究.本文是系列文章的首篇,在本文中我们搜集了87年以前的全部、及87年以后的大部分中、高分辨率CIV吸收线光谱资料,给出了包含228个类星体的均匀无偏统计样本,该样本有CIV吸收线229条,系由228个类星体的所有534条吸收线中选出.此外,我们还定义了4个子样本.     

CuNi-VTiFe复合型矿化镁铁-超镁铁杂岩体岩相学及岩石地球化学特征:以新疆北部为例

铜镍硫化物矿床和钒钛磁铁矿矿床是镁铁-超镁铁杂岩重要的矿床类型,但二者共生的情况在国内还不多见。新疆北部这类铜镍-钒钛铁复合型矿化岩体较为发育,目前已发现有香山、牛毛泉、土墩南和哈拉达拉等4个岩体属于此类。它们的成岩时代多集中在早二叠世,出露面积在2.8～22km~2,介于通道型铜镍矿化小岩体和大型层状岩体之间,韵律构造发育;岩石组合为超基性-基性-中性岩类,以出现浅色的闪长岩或淡色辉长岩为特点,岩石中金属矿物氧化物(钛铁矿、磁铁矿)和硫化物(黄铁矿、磁黄铁矿、黄铜矿,有时有镍黄铁矿)共存和共生;含矿岩石组合和岩石化学特征与典型铜镍硫化物矿床和钒钛磁铁矿矿床相比,具有重叠和过渡特征;稀土和微量元素特征反映出杂岩体不同岩石类型可能具有相同或相似岩浆来源,是经过强烈分异和演化的产物。新疆北部这类复合型矿化,与北疆地区典型铜镍矿床和典型钒钛磁铁矿矿床,共同构成了新疆北部后碰撞幔源岩浆矿床成矿谱系。     

Ensemble level upscaling for compositional flow simulation

Uncertainty quantification is typically accomplished by simulating multiple geological realizations, which can be very expensive computationally if the flow process is complicated and the models are highly resolved. Upscaling procedures can be applied to reduce computational demands, though it is essential that the resulting coarse-model predictions correspond to reference fine-scale solutions. In this work, we develop an ensemble level upscaling (EnLU) procedure for compositional systems, which enables the efficient generation of multiple coarse models for use in uncertainty quantification. We apply a newly developed global compositional upscaling method to provide coarse-scale parameters and functions for selected realizations. This global upscaling entails transmissibility and relative permeability upscaling, along with the computation of a-factors to capture component fluxes. Additional features include near-well upscaling for all coarse parameters and functions, and iteration on the a-factors, which is shown to improve accuracy. In the EnLU framework, this global upscaling is applied for only a few selected realizations. For 90 % or more of the realizations, upscaled functions are assigned statistically based on quickly computed flow and permeability attributes. A sequential Gaussian co-simulation procedure is incorporated to provide coarse models that honor the spatial correlation structure of the upscaled properties. The resulting EnLU procedure is applied for multiple realizations of two-dimensional models, for both Gaussian and channelized permeability fields. Results demonstrate that EnLU provides P10, P50, and P90 results for phase and component production rates that are in close agreement with reference fine-scale results. Less accuracy is observed in realization-by-realization comparisons, though the models are still much more accurate than those generated using standard coarsening procedures.