Upscaled modeling of well singularity for simulating flow in heterogeneous formations

Subsurface flows are affected by geological variability over a range of length scales. The modeling of well singularity in heterogeneous formations is important for simulating flow in aquifers and petroleum reservoirs. In this paper, two approaches in calculating the upscaled well index to capture the effects of fine scale heterogeneity in near-well regions are presented and applied. We first develop a flow-based near-well upscaling procedure for geometrically flexible grids. This approach entails solving local well-driven flows and requires the treatment of geometric effects due to the nonalignment between fine and coarse scale grids. An approximate coarse scale well model based on a well singularity analysis is also proposed. This model, referred to as near-well arithmetic averaging, uses only the fine scale permeabilities at well locations to compute the coarse scale well index; it does not require solving any flow problems. These two methods are systematically tested on three-dimensional models with a variety of permeability distributions. It is shown that both approaches provide considerable improvement over a simple (arithmetic) averaging approach to compute the coarse scale well index. The flow-based approach shows close agreement to the fine scale reference model, and the near-well arithmetic averaging also offers accuracy for an appropriate range of parameters. The interaction between global flow and near-well upscaling is also investigated through the use of global fine scale solutions in near-well scale-up calculations.     

Permeability Upscaling Measured on a Block of Berea Sandstone: Results and Interpretation

To physically investigate permeability upscaling, over 13,000 permeability values were measured with four different sample supports (i.e., sample volumes) on a block of Berea Sandstone. At each sample support, spatially exhaustive permeability datasets were measured, subject to consistent flow geometry and boundary conditions, with a specially adapted minipermeameter test system. Here, we present and analyze a subset of the data consisting of 2304 permeability values collected from a single block face oriented normal to stratification. Results reveal a number of distinct and consistent trends (i.e., upscaling) relating changes in key summary statistics to an increasing sample support. Examples include the sample mean and semivariogram range that increase with increasing sample support and the sample variance that decreases. To help interpret the measured mean upscaling, we compared it to theoretical models that are only available for somewhat different flow geometries. The comparison suggests that the nonuniform flow imposed by the minipermeameter coupled with permeability anisotropy at the scale of the local support (i.e., smallest sample support for which data is available) are the primary controls on the measured upscaling. This work demonstrates, experimentally, that it is not always appropriate to treat the local-support permeability as an intrinsic feature of the porous medium, that is, independent of its conditions of measurement.     

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.     

Modified adaptive local–global upscaling method for discontinuous permeability distribution

The paper is devoted to the upscaling method appropriate for single-phase flow in media with discontinuous permeability distribution. The suggested algorithm is a modification of the iterative adaptive local–global upscaling developed by Chen and coauthors. The key feature of this method is a consistency between local and coarse global calculated characteristics. In this work, we apply a modified procedure to determine the boundary conditions used in the local fine-scale computation. To increase the accuracy of these boundary conditions on each iteration, we involve an additional preliminary step based on the results of coarse scale calculations from the previous iteration. Numerical tests show an essential improvement of the accuracy of upscaled flow rates for most of the realizations of statistical permeability distribution. Although the developed method is universal, its efficiency increases with increasing of permeability contrast.     

Statistical assignment of upscaled flow functions for an ensemble of geological models

Upscaled flow functions are often needed to account for the effects of fine-scale permeability heterogeneity in coarse-scale simulation models. We present procedures in which the required coarse-scale flow functions are statistically assigned to an ensemble of upscaled geological models. This can be viewed as an extension and further development of a recently developed ensemble level upscaling (EnLU) approach. The method aims to efficiently generate coarse-scale flow models capable of reproducing the ensemble statistics (e.g., cumulative distribution function) of fine-scale flow predictions for multiple reservoir models. The most expensive part of standard coarsening procedures is typically the generation of upscaled two-phase flow functions (e.g., relative permeabilities). EnLU provides a means for efficiently generating these upscaled functions using stochastic simulation. This involves the use of coarse-block attributes that are both fast to compute and correlate closely with the upscaled two-phase functions. In this paper, improved attributes for use in EnLU, namely the coefficient of variation of the fine-scale single-phase velocity field (computed during computation of upscaled absolute permeability) and the integral range of the fine-scale permeability variogram, are identified. Geostatistical simulation methods, which account for spatial correlations of the statistically generated upscaled functions, are also applied. The overall methodology thus enables the efficient generation of coarse-scale flow models. The procedure is tested on 3D well-driven flow problems with different permeability distributions and variable fluid mobility ratios. EnLU is shown to capture the ensemble statistics of fine-scale flow results (water and oil flow rates as a function of time) with similar accuracy to full flow-based upscaling methods but with computational speedups of more than an order of magnitude.     

Upscaling Hydraulic Conductivities in Cross-Bedded Formations

Modern geostatistical techniques allow the generation of high-resolution heterogeneous models of hydraulic conductivity containing millions to billions of cells. Selective upscaling is a numerical approach for the change of scale of fine-scale hydraulic conductivity models into coarser scale models that are suitable for numerical simulations of groundwater flow and mass transport. Selective upscaling uses an elastic gridding technique to selectively determine the geometry of the coarse grid by an iterative procedure. The geometry of the coarse grid is built so that the variances of flow velocities within the coarse blocks are minimum. Selective upscaling is able to handle complex geological formations and flow patterns, and provides full hydraulic conductivity tensor for each block. Selective upscaling is applied to a cross-bedded formation in which the fine-scale hydraulic conductivities are full tensors with principal directions not parallel to the statistical anisotropy of their spatial distribution. Mass transport results from three coarse-scale models constructed by different upscaling techniques are compared to the fine-scale results for different flow conditions. Selective upscaling provides coarse grids in which mass transport simulation is in good agreement with the fine-scale simulations, and consistently superior to simulations on traditional regular (equal-sized) grids or elastic grids built without accounting for flow velocities.     

Upscaling permeability for fractured concrete: meso–macro numerical approach coupled to strong discontinuities

A two‐scales numerical analysis is set up in order to upscale the permeability of fractured materials such as concrete. To that aim, we couple finite element (FE) kinematics enhancements (strong discontinuities) representing fine scale cracks to the fine scale permeability tensor. The latter may be split into two parts: the first one is isotropic and corresponds to flows within the porosity of the material; the second one, based upon a set of cracks with different orientations and openings, is anisotropic. For the latter, each crack is a path for mass flow according to the Poiseuille law considering two infinite planes. We show how the upscaling procedure leads both to the definition of macroscopic permeability tensors as well as the flow rate evaluation for components of concrete structures. Copyright © 2013 John Wiley & Sons, Ltd.     

空间尺度转换与跨尺度信息链接:区域生态水文模拟研究空间尺度转换方法综述

空间尺度转换是近年来区域生态水文研究领域的一个基本研究问题。其需要主要是源于模型的输入数据与所能提供的数据空间尺度不一致以及模型所代表的地表过程空间尺度与所观测的地表过程空间尺度不吻合。综述了目前区域生态水文模拟研究中常用的空间尺度转换研究方法,包括向上尺度转换和向下尺度转换。详细论述了2种向下尺度转换方法: 统计学经验模型和动态模型。前者是通过将GCM大尺度数据与长期的历史观测数据比较从而建立统计学相关模型, 然后利用这个统计学经验模型进行向下的空间尺度转换. 然而动态模型并不直接对GCM数据进行向下尺度的转换,而是对与GCM进行动态耦合的区域气候模型(RCM) 的输出数据进行空间尺度转换. 通常后者所获得的数据精度要比前者高,但是一个主要缺点就是并不是全球所有的研究区域都有对应的RCM。还详细论述了2种向上尺度转换方法: 统计学经验模型和斑块模型。前者是建立一个能代表小尺度信息在大尺度上分布的密度分布概率函数, 然后利用这个函数在所需的大尺度上进行积分而求得大尺度所需的信息。而后者是根据相似性最大化原则将大尺度划分为若干个可操作的小尺度斑块,然后将计算的每个小尺度斑块的信息平均化得到大尺度所需的信息。通常在计算这种斑块化的小尺度信息的时候,对每个小尺度也会采用统计学经验模型来计算代表整个斑块小尺度的信息。建议用斑块模型与统计学经验模型相集合的方法来实现向上的空间尺度转换     

Multiscale simulations of porous media flows in flow-based coordinate system

In this paper, we propose a multiscale technique for the simulation of porous media flows in a flow-based coordinate system. A flow-based coordinate system allows us to simplify the scale interaction and derive the upscaled equations for purely hyperbolic transport equations. We discuss the applications of the method to two-phase flows in heterogeneous porous media. For two-phase flow simulations, the use of a flow-based coordinate system requires limited global information, such as the solution of single-phase flow. Numerical results show that one can achieve accurate upscaling results using a flow-based coordinate system.     

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.     

不同地质条件约束下的油气资源量概率组合加和方法及应用

目前, 国内普遍使用的“简单加和”或“概率加权加和”方法计算的区带或三(四)级圈闭资源量很大程度上与实际地质情况并不吻合.在PetroV开发过程中, 设计并实现了一种可充分考虑不同地质条件约束的油气资源量计算方法——以不确定性体积法的随机蒙氏模拟为基础, 对最新地质上识别出的、不可切分的含油气聚集单元, 进行符合不同地质模型约束的取样、组合; 在明确了哪些单元才应该同时出现的基础上, 将其他不同地质条件约束有机融合到模拟过程中, 客观描述各个计算单元体积模型的不确定性.从实例计算结果分析可以看出, “概率组合加和”可获取更加符合当前地质模型的不确定性油气资源量计算结果, 显著提升勘探部署项目优选排队的合理性.      

Identifying the Representative Elementary Volume for Permeability in Heterolithic Deposits Using Numerical Rock Models

Using a range of realistic 3D numerical lithofacies (dm-scale) models of ripple laminated sandstone intercalated with mudstone we evaluate how single-phase permeability varies as a function of sample support. The models represent a range of mudstone content which is typical for tidal deposits. Furthermore, the spatial distribution of flow barriers (i.e. mudstone) is not random, but governed by sedimentological rules giving a variable anisotropy ratio as a function of mudstone content. Both vertical and horizontal permeability are found to vary at small sample volumes, but these fluctuations reduce as the sample volume increases. The vertical permeability increases while the horizontal permeability is nearly constant as a function of sample support for small mudstone contents. For higher mudstone content, the horizontal permeability decreases while the vertical permeability is nearly constant as a function of sample support. We propose a criterion, based on a normalised standard deviation, to determine the Representative Elementary Volume (REV). The size of the REV is dependent on both the property measured (vertical and horizontal permeability) and the correlation lengths of the lithological elements (i.e. lithofacies). Based on this we identify three flow upscaling regimes that each require a different method for upscaling: (1) layered systems where the arithmetic and harmonic averages are appropriate, (2) systems close to the percolation threshold where a percolation model should be used, and (3) discontinuous systems where an effective medium method provides the best estimate of permeability. The work gives, by using numerical experiments on a range of heterogeneous systems, a new insight in determination of the REV for permeability at the lithofacies scale and its relation to sedimentological parameters.     

基于Monte Carlo模拟的三维剖面地质界线不确定性分析

三维剖面地质界线是构建三维地质结构模型的重要基础数据,其不确定性会影响三维模型的几何形态和属性分布。以单一分布为假设前提的统计学不确定性分析方法掩盖了其他概率分布特征对模型的影响。突破单一误差分布条件的假设前提,本文使用Monte Carlo方法模拟了不同概率分布情况下地质剖面数据中地质界线的抽样采集,以及地质界线空间分布的不确定性;依托地质界线空间位置与地质属性的耦合关系,提出了用地质属性概率分布实现地质界线空间不确定性的定量可视化,并结合实际地质剖面探讨了多种概率分布条件下地质界线的空间不确定性。实例研究表明,基于Monte Carlo模拟的不确定性分析方法可以突破单一误差分布假设条件,结合地质属性概率可充分揭示出建模数据的内在不确定性与模型外在要素形态之间的耦合关系。     

Upscaling Uncertain Permeability Using Small Cell Renormalization

Sedimentary rocks have structures on all length scales from the millimeter to the kilometer. These structures are generally associated with variations in rock permeability. These need to be modeled if we are to make predictions about fluid flow through the rock. However, existing computers are not powerful enough for us to be able to represent all scales of heterogeneity explicitly in our fluid flow models—hence, we need to upscale. Small cell renormalization is a fast method for upscaling permeability, derived from an analogue circuit of resistors. However, it assumes that the small scale permeability distribution is known. In practice, this is unlikely. The only information available about small scale properties is either qualitative, derived from the depositional setting of the reservoir, or local to the wells as a result of coring or logging. The influence of small scale uncertainty on large scale properties is usually modelled by the Monte Carlo method. This is time-consuming and inaccurate if not enough realisations are used. This paper describes a new implementation of renormalization, which enables the direct upscaling of uncertain small-scale permeabilities to produce the statistical properties of the equivalent coarse grid. This is achieved by using a perturbation expansion of the resistor-derived equation. The method is verified by comparison with numerical simulations using the Monte Carlo method. The prediction of expected large-scale permeability and its standard deviation are shown to be accurate for small cell standard deviations of up to 40% of the mean cell value, using just the first nonzero term of the perturbation expansion. Inclusion of higher order terms allows larger standard deviations to be modeled accurately. Evaluation of cross-terms allows correlations of actual cell values, over and above the background structure of mean cell values. The perturbation method is significantly faster than conventional Monte Carlo simulation. It needs just two calculations whereas the Monte Carlo method needs many thousands of realisations to be generated and renormalized to converge. This results in significant savings in computer time.     

论不连序系列样本概率权重矩不偏计算公式

人们普遍认为要从理论上通过数学分析的方法来论证不连序系列样本概率权重矩的偏倚性是很困难的。本文根据次序统计量理论，分析推证了不连序系列样本概率权重矩的不偏计算公式，论证结果具有较重要的理论和实用意义。     

A coarse-scale compositional model

In subsurface flow modeling, compositional simulation is often required to model complex recovery processes, such as gas/CO ₂ injection. However, compositional simulation on fine-scale geological models is still computationally expensive and even prohibitive. Most existing upscaling techniques focus on black-oil models. In this paper, we present a general framework to upscale two-phase multicomponent flow in compositional simulation. Unlike previous studies, our approach explicitly considers the upscaling of flow and thermodynamics. In the flow part, we introduce a new set of upscaled flow functions that account for the effects of compressibility. This is often ignored in the upscaling of black-oil models. In the upscaling of thermodynamics, we show that the oil and gas phases within a coarse block are not at chemical equilibrium. This non-equilibrium behavior is modeled by upscaled thermodynamic functions, which measure the difference between component fugacities among the oil and gas phases. We apply the approach to various gas injection problems with different compositional features, permeability heterogeneity, and coarsening ratios. It is shown that the proposed method accurately reproduces the averaged fine-scale solutions, such as component overall compositions, gas saturation, and density solutions in the compositional flow.     

Application of multiple-point geostatistics on modelling groundwater flow and transport in a cross-bedded aquifer (Belgium)

Sedimentological processes often result in complex three-dimensional subsurface heterogeneity of hydrogeological parameter values. Variogram-based stochastic approaches are often not able to describe heterogeneity in such complex geological environments. This work shows how multiple-point geostatistics can be applied in a realistic hydrogeological application to determine the impact of complex geological heterogeneity on groundwater flow and transport. The approach is applied to a real aquifer in Belgium that exhibits a complex sedimentary heterogeneity and anisotropy. A training image is constructed based on geological and hydrogeological field data. Multiple-point statistics are borrowed from this training image to simulate hydrofacies occurrence, while intrafacies permeability variability is simulated using conventional variogram-based geostatistical methods. The simulated hydraulic conductivity realizations are used as input to a groundwater flow and transport model to investigate the effect of small-scale sedimentary heterogeneity on contaminant plume migration. Results show that small-scale sedimentary heterogeneity has a significant effect on contaminant transport in the studied aquifer. The uncertainty on the spatial facies distribution and intrafacies hydraulic conductivity distribution results in a significant uncertainty on the calculated concentration distribution. Comparison with standard variogram-based techniques shows that multiple-point geostatistics allow better reproduction of irregularly shaped low-permeability clay drapes that influence solute transport.