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.     

Sub-pixel Mapping of Coarse Satellite Remote Sensing Images with Stochastic Simulations from Training Images

Super-resolution or sub-pixel mapping is the process of providing fine scale land cover maps from coarse-scale satellite sensor information. Such a procedure calls for a prior model depicting the spatial structures of the land cover types. When available, an analog of the underlying scene (a training image) may be used for such a model. The single normal equation simulation algorithm (SNESIM) allows extracting the relevant pattern information from the training image and uses that information to downscale the coarse fraction data into a simulated fine scale land cover scene. Two non-exclusive approaches are considered to use training images for super-resolution mapping. The first one downscales the coarse fractions into fine-scale pre-posterior probabilities which is then merged with a probability lifted from the training image. The second approach pre-classifies the fine scale patterns of the training image into a few partition classes based on their coarse fractions. All patterns within a partition class are recorded by a search tree; there is one tree per partition class. At each fine scale pixel along the simulation path, the coarse fraction data is retrieved first and used to select the appropriate search tree. That search tree contains the patterns relevant to that coarse fraction data. To ensure exact reproduction of the coarse fractions, a servo-system keeps track of the number of simulated classes inside each coarse fraction. Being an under-determined stochastic inverse problem, one can generate several super resolution maps and explore the space of uncertainty for the fine scale land cover. The proposed SNESIM sub-pixel resolution mapping algorithms allow to: (i) exactly reproduce the coarse fraction, (ii) inject the structural model carried by the training image, and (iii) condition to any available fine scale ground observations. Two case studies are provided to illustrate the proposed methodology using Landsat TM data from southeast China.     

Integrating Seismic Data for Lithofacies Modeling: A Comparison of Sequential Indicator Simulation Algorithms

Clastic reservoir characterization starts typically with modeling lithofacies distribution and geometry. The architecture of the reservoir, governed by the lithofacies geometry, is a major source of heterogeneity in such clastic systems. Seismic data provide potentially valuable information about the areal distribution of different lithofacies, such as the averaged prior proportion of each lithofacies. However, seismic data are available only at coarse vertical resolution rather than the fine lithofacies sampling along wells, hence seismic is considered equivalent to 2D data while building 3D geological models. This scale difference between the seismic data and the lithofacies data available along the wells makes direct integration difficult. Different algorithms have been proposed to integrate the seismic data: (1) duplicate seismic data along the vertical line and use the prior proportions provided by the seismic data as prior local means; (2) integrate the 2D seismic data as collocated block averages; and (3) duplicate seismic data along the vertical line and integrate them using a Markov-Bayes algorithm. These three algorithms are applied on a data set originating from a real clastic reservoir. The results are compared with regard to how much kriging weight is applied to the seismic data and how well the information from seismic data is honored.     

Nested gridding and streamline-based simulation for fast reservoir performance prediction

Detailed reservoir models routinely contain 10⁶–10⁸ grid blocks. These models often cannot be used directly in a reservoir simulation because of the time and memory required for solving the pressure grid on the fine grid. We propose a nested gridding technique that efficiently obtains an approximate solution for the pressure field. The domain is divided into a series of coarse blocks, each containing several fine cells. Effective mobilities are computed for each coarse grid block and the pressure is then found on the coarse scale. The pressure field within each coarse block is computed using flux boundary conditions obtained from the coarse pressure solution. Streamline-based simulation is used to move saturations forward in time. We test the method for a series of example waterflood problems and demonstrate that the method can give accurate estimates of oil production for large 3D models significantly faster than direct simulation using streamlines on the fine grid, making the method overall approximately up to 1,000 times faster than direct conventional simulation.     

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.     

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。     

Multiscale mixed/mimetic methods on corner-point grids

Multiscale simulation is a promising approach to facilitate direct simulation of large and complex grid models for highly heterogeneous petroleum reservoirs. Unlike traditional simulation, approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating subscale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (fine-scale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderately sized) coarse grid, one can retain the efficiency of an upscaling method and, at the same time, produce detailed and conservative velocity fields on the underlying fine grid. For pressure equations, the multiscale mixed finite-element method (MsMFEM) has been shown to be a particularly versatile approach. In this paper, we extend the method to corner-point grids, which is the industry standard for modelling complex reservoir geology. To implement MsMFEM, one needs a discretisation method for solving local flow problems on the underlying fine grids. In principle, any stable and conservative method can be used. Here, we use a mimetic discretisation, which is a generalisation of mixed finite elements that gives a discrete inner product, allows for polyhedral elements, and can (easily) be extended to curved grid faces. The coarse grid can, in principle, be any partition of the subgrid, where each coarse block is a connected collection of subgrid cells. However, we argue that, when generating coarse grids, one should follow certain simple guidelines to achieve improved accuracy. We discuss partitioning in both index space and physical space and suggest simple processing techniques. The versatility and accuracy of the new multiscale mixed methodology is demonstrated on two corner-point models: a small Y-shaped sector model and a complex model of a layered sedimentary bed. A variety of coarse grids, both violating and obeying the above mentioned guidelines, are employed. The MsMFEM solutions are compared with a reference solution obtained by direct simulation on the subgrid.     

Maximum likelihood classification for facies inference from reservoir attributes

A cluster analysis methodology is developed to recover facies realizations from observed reservoir attributes. A maximum likelihood estimator allows us for identifying the most probable underlying facies using a spatial clustering algorithm. In seismic characterization, this algorithm can yield relevant geological models for subsequent history-matching studies. In history-matching procedures, it provides informative facies maps as well as starting points for further studies.     

考虑页岩纹层与裂缝网络的延长组页岩多尺度三维地质结构模型

现场调查表明,砂质纹层、凝灰质纹层和天然裂缝广泛地存在于陆相页岩储层中。本文对鄂尔多斯盆地页岩储层中的纹层和天然裂缝进行了多尺度研究,并构建了三维地质结构模型。首先,基于二维裂缝现场调查,利用蒙特卡罗模拟方法建立了研究区域的三维裂缝网络模型。然后通过多种观测手段获得由宏观尺度到微观尺度的纹层结构特征。对多尺度纹层厚度的统计分析表明,米级、分米级、厘米级、毫米级和10微米级等不同研究尺度下的纹层平均厚度分别为2.26 m,2.09 dm,1.70 cm,1.48 mm和11.7 μm,呈现出分形特征,分形维数为1.06;不同研究尺度下的单层厚度均服从负指数分布规律,即各研究尺度下厚度越大的纹层,其层数越少,反之越薄的纹层其数量越多。最后,根据上述纹层平均厚度及概率分布函数特征,建立了页岩的多尺度纹层结构模型,并将其叠加在裂缝网络模型上,生成不同尺度下的页岩三维地质结构模型。模型输出的裂缝、纹层参数与研究区域的真实地质参数有着较好的对比验证。这项研究工作可为页岩气储层的水力压裂数值模拟和物理模型试验提供更可靠的地质模型。     

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.     

Identifying influence areas with connectivity analysis – application to the local perturbation of heterogeneity distribution for history matching

Numerical representations of a target reservoir can help to assess the potential of different development plans. To be as predictive as possible, these representations or models must reproduce the data (static, dynamic) collected on the field. However, constraining reservoir models to dynamic data – the history-matching process – can be very time consuming. Many uncertain parameters need to be taken into account, such as the spatial distribution of petrophysical properties. This distribution is mostly unknown and usually represented by millions of values populating the reservoir grid. Dedicated parameterization techniques make it possible to investigate many spatial distributions from a small number of parameters. The efficiency of the matching process can be improved from the perturbation of specific regions of the reservoir. Distinct approaches can be considered to define such regions. For instance, one can refer to streamlines. The leading idea is to identify areas that influence the production behavior where the data are poorly reproduced. Here, we propose alternative methods based on connectivity analysis to easily provide approximate influence areas for any fluid-flow simulation. The reservoir is viewed as a set of nodes connected by weighted links that characterize the distance between two nodes. The path between nodes (or grid blocks) with the lowest cumulative weight yields an approximate flow path used to define influence areas. The potential of the approach is demonstrated on the basis of 2D synthetic cases for the joint integration of production and 4D saturation data, considering several formulations for the weights attributed to the links.     

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.     

Generalized Coarse Graining Procedures for Flow in Porous Media

This paper focuses on heterogeneous soil conductivities and on the impact their resolution has on a solution of the piezometric head equation: owing to spatial variations of the conductivity, the flow properties at larger scales differ from those found for experiments performed at smaller scales. The method of coarse graining is proposed in order to upscale the piezometric head equation on arbitrary intermediate scales. At intermediate scales large scale fluctuations of the conductivities are resolved, whereas small scale fluctuations are smoothed by a partialy spatial filtering procedure. The filtering procedure is performed in Fourier space with the aid of a low-frequency cut-off function. We derive the partially upscaled head equations. In these equations, the impact of the small scale variability is modeled by scale dependent effective conductivities which are determined by additional differential equations. Explicit results for the scale dependent conductivity values are presented in lowest order perturbation theory. The perturbation theory contributions are summed up with using a renormalisation group analysis yielding explicit results for the effective conductivity in isotropic media. Therefore, the results are also valid for highly heterogeneous media. The results are compared with numerical simulations performed by Dykaar and Kitanidis (1992). The method of coarse graining combined by a renormalisation group analysis offers a tool to derive exact and explicit expressions for resolution dependent conductivity values. It is, e.g., relevant for the interpretation of measurement data on different scales and for reduction of grid-block resolution in numerical modeling. This revised version was published online in July 2006 with corrections to the Cover Date.     

多信息关联的辫状河储层夹层预测方法研究:以南苏丹P油田Fal块为例

辫状河储层的夹层预测是油藏描述的重点内容。目前夹层的预测主要集中于夹层发育模式研究和心滩坝体的构型单元解剖,且多运用单一的预测方法。南苏丹P油田辫状河储层夹层类型多、规模差异大、分布复杂,定量表征难度较大,在文献调研的基础上,从夹层的沉积成因入手,依据不同沉积方式形成的沉积砂体及其内部泥质夹层形态与结构不同的特点,综合岩心、测井与地震等多种资料,提出多信息关联的辫状河储层夹层预测方法。在密井网区建立骨架剖面与三角网小剖面,运用测井资料的垂向高分辨率与地震资料的横向强连续性特征确定不同类型夹层的井间发育规模;在建立岩相模型的基础上,以隔层厚度分布图为约束条件,采用确定性建模方法建立稳定泥岩隔层分布模型;以沉积微相研究结果和夹层规模预测结果为约束条件,采用随机建模方法分别在砂岩相和泥岩非隔层相中模拟心滩坝、河道和各类型夹层的分布;最终确定了研究区主要存在4种成因类型的夹层,并在多信息关联的基础上建立反映多类型夹层空间分布的辫状河储层精细地质模型。研究发现,对于厚度大于2 m的夹层可以通过井震结合的方法验证其井间规模,定量确定不同层位、不同类型夹层顺物源与切物源的发育规模,为夹层模型的建立奠定基础;基于克里金插值方法建立的岩相概率模型增加岩相模型准确率至94%;以隔层厚度平面分布图为约束条件的确定性建模方法可准确建立砂组及小层间隔层分布模型;在各成因类型夹层井间规模预测的基础上,基于目标的随机模拟方法可以针对不同成因类型夹层的发育形态、数量、规模和趋势分别设定模拟参数,确定性与随机性相结合,实现了辫状河储层精细地质模型的建立。同时,对相关储层的夹层预测具有一定的指导作用。     

FV-MHMM method for reservoir modeling

The present paper proposes a new family of multiscale finite volume methods. These methods usually deal with a dual mesh resolution, where the pressure field is solved on a coarse mesh, while the saturation fields, which may have discontinuities, are solved on a finer reservoir grid, on which petrophysical heterogeneities are defined. Unfortunately, the efficiency of dual mesh methods is strongly related to the definition of up-gridding and down-gridding steps, allowing defining accurately pressure and saturation fields on both fine and coarse meshes and the ability of the approach to be parallelized. In the new dual mesh formulation we developed, the pressure is solved on a coarse grid using a new hybrid formulation of the parabolic problem. This type of multiscale method for pressure equation called multiscale hybrid-mixed method (MHMM) has been recently proposed for finite elements and mixed-finite element approach (Harder et al. 2013). We extend here the MH-mixed method to a finite volume discretization, in order to deal with large multiphase reservoir models. The pressure solution is obtained by solving a hybrid form of the pressure problem on the coarse mesh, for which unknowns are fluxes defined on the coarse mesh faces. Basis flux functions are defined through the resolution of a local finite volume problem, which accounts for local heterogeneity, whereas pressure continuity between cells is weakly imposed through flux basis functions, regarded as Lagrange multipliers. Such an approach is conservative both on the coarse and local scales and can be easily parallelized, which is an advantage compared to other existing finite volume multiscale approaches. It has also a high flexibility to refine the coarse discretization just by refinement of the lagrange multiplier space defined on the coarse faces without changing nor the coarse nor the fine meshes. This refinement can also be done adaptively w.r.t. a posteriori error estimators. The method is applied to single phase (well-testing) and multiphase flow in heterogeneous porous media.     

Use of Border Regions for Improved Permeability Upscaling

A procedure for the improved calculation of upscaled grid block permeability tensors on Cartesian grids is described and applied. The method entails the use of a border region of fine-scale cells surrounding the coarse block for which the upscaled permeability is to be computed. The implementation allows for the use of full-tensor permeability fields on the fine and coarse scales. Either periodic or pressure–no flow boundary conditions are imposed over the extended local domain (target block plus border regions) though averaged quantities, used to compute the upscaled permeability tensor, are computed only over the target block region. Flow and transport results using this procedure are compared to those from standard methods for different types of geological and simulation models. Improvement using the new approach is consistently observed for the cases considered, though the degree of improvement varies for different models and flow quantities.     

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.     

高精度层序地层学和储层预测

当前层序地层学的研究不断从盆地规模的层序地层和体系域分析向储层规模的高精度层序地层学的方向深化。层序地层学的概念和方法可应用于从盆地到储层的各种规模的沉积充填分析。高精度层序地层学是以露头、岩芯、测井和高分辨地震等密集控制的资料分析为基础的。精细的测井分析、高分辨三维地震剖面和各种参数处理和切片技术、计算机模拟及可视化技术等是开展高精度层序地层学研究和应用于地下沉积地质分析的重要支持。高精度层序地层学的概念和方法为盆地沉积充填的精细研究、储集体分布和储层不均一性预测以及开发地质等研究提供了重要的方法和手段