Combining the modified discrete element method with the virtual element method for fracturing of porous media

Simulation of fracturing processes in porous rocks can be divided into two main branches: (i) modeling the rock as a continuum enhanced with special features to account for fractures or (ii) modeling the rock by a discrete (or discontinuous) approach that describes the material directly as a collection of separate blocks or particles, e.g., as in the discrete element method (DEM). In the modified discrete element (MDEM) method, the effective forces between virtual particles are modified so that they reproduce the discretization of a first-order finite element method (FEM) for linear elasticity. This provides an expression of the virtual forces in terms of general Hook’s macro-parameters. Previously, MDEM has been formulated through an analogy with linear elements for FEM. We show the connection between MDEM and the virtual element method (VEM), which is a generalization of FEM to polyhedral grids. Unlike standard FEM, which computes strain-states in a reference space, MDEM and VEM compute stress-states directly in real space. This connection leads us to a new derivation of the MDEM method. Moreover, it enables a direct coupling between (M)DEM and domains modeled by a grid made of polyhedral cells. Thus, this approach makes it possible to combine fine-scale (M)DEM behavior near the fracturing region with linear elasticity on complex reservoir grids in the far-field region without regridding. To demonstrate the simulation of hydraulic fracturing, the coupled (M)DEM-VEM method is implemented using the Matlab Reservoir Simulation Toolbox (MRST) and linked to an industry-standard reservoir simulator. Similar approaches have been presented previously using standard FEM, but due to the similarities in the approaches of VEM and MDEM, our work provides a more uniform approach and extends these previous works to general polyhedral grids for the non-fracturing domain.     

Comparison between cell-centered and nodal-based discretization schemes for linear elasticity

In this paper, we study newly developed methods for linear elasticity on polyhedral meshes. Our emphasis is on applications of the methods to geological models. Models of subsurface, and in particular sedimentary rocks, naturally lead to general polyhedral meshes. Numerical methods which can directly handle such representation are highly desirable. Many of the numerical challenges in simulation of subsurface applications come from the lack of robustness and accuracy of numerical methods in the case of highly distorted grids. In this paper, we investigate and compare the Multi-Point Stress Approximation (MPSA) and the Virtual Element Method (VEM) with regard to grid features that are frequently seen in geological models and likely to lead to a lack of accuracy of the methods. In particular, we look at how the methods perform near the incompressible limit. This work shows that both methods are promising for flexible modeling of subsurface mechanics.     

A virtual element method for the two-phase flow of immiscible fluids in porous media

A primal C⁰-conforming virtual element discretization for the approximation of the bidimensional two-phase flow of immiscible fluids in porous media using general polygonal meshes is discussed. This work investigates the potentialities of the Virtual Element Method (VEM) in solving this specific problem of immiscible fluids in porous media involving a time-dependent coupled system of non-linear partial differential equations. The performance of the fully discrete scheme is thoroughly analysed testing it on general meshes considering both a regular problem and more realistic benchmark problems that are of interest for physical and engineering applications.

     

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.     

Algebraic multigrid techniques for discontinuous Galerkin methods with varying polynomial order

We present a parallel algebraic multigrid (AMG) algorithm for the implicit solution of the Darcy problem discretized by the discontinuous Galerkin (DG) method that scales optimally for regular and irregular meshes. The main idea centers on recasting the preconditioning problem so that existing AMG solvers for nodal lower order finite elements can be leveraged. This is accomplished by a transformation operator which maps the solution from a Lagrange basis representation to a Legendre basis representation. While this mapping function must be user supplied, we demonstrate how easily it can be constructed for somepopular finite element representations includingquadrilateral/hexahedral and triangular/tetrahedral DG formulations. Furthermore, we show that the mapping does not depend on the Jacobian transformation between reference and physical space and so it can be constructed with very limited mesh information. Parallel performance studies demonstrate the versatility of this approach.     

Fabric evolution within shear bands of granular materials and its relation to critical state theory

In an effort to study the relation of fabrics to the critical states of granular aggregates, the discrete element method (DEM) is used to investigate the evolution of fabrics of virtual granular materials consisting of 2D elongated particles. Specimens with a great variety of initial fabrics in terms of void ratios, preferred particle orientations, and intensities of fabric anisotropy were fabricated and tested with direct shear and biaxial compression tests. During loading of a typical specimen, deformation naturally localizes within shear bands while the remaining of the sample stops deforming. Thus, studying the evolution of fabric requires performing continuous local fabric measurements inside these bands, a suitable task for the proposed DEM methodology. It is found that a common ultimate/critical state is eventually reached by all specimens regardless of their initial states. The ultimate/critical state is characterized by a critical void ratio e which depends on the mean stress p, while the other critical state fabric variables related to particle orientations are largely independent of p. These findings confirm the uniqueness of the critical state line in the e ? p space, and show that the critical state itself is necessarily anisotropic. Additional findings include the following: (1) shear bands are highly heterogeneous and critical states exist only in a statistical sense; (2) critical states can only be reached at very large local shear deformations, which are not always obtained by biaxial compression tests (both physical and numerical); (3) the fabric evolution processes are very complex and highly dependent on the initial fabrics. Copyright © 2010 John Wiley & Sons, Ltd.     

Mortar Upscaling for Multiphase Flow in Porous Media

In mortar space upscaling methods, a reservoir is decomposed into a series of subdomains (blocks) in which independently constructed numerical grids and possibly different physical models and discretization techniques can be employed in each block. Physically meaningful matching conditions are imposed on block interfaces in a numerically stable and accurate way using mortar finite element spaces. Coarse mortar grids and fine subdomain grids provide two-scale approximations. In the resulting effective solution flow is computed in subdomains on the fine scale while fluxes are matched on the coarse scale. In addition the flexibility to vary adaptively the number of interface degrees of freedom leads to more accurate multiscale approximations. This methodology has been implemented in the Center for Subsurface Modeling's multiphysics multiblock simulator IPARS (Integrated Parallel Accurate reservoir Simulator). Computational experiments demonstrate that this approach is scalable in parallel and it can be applied to non-matching grids across the interface, multinumerics and multiphysics models, and mortar adaptivity. Moreover unlike most upscaling approaches the underlying systems can be treated fully implicitly.     

A fully implicit and consistent finite element framework for modeling reservoir compaction with large deformation and nonlinear flow model. Part II: verification and numerical example

A fully implicit, fully coupled, and fully consistent finite element framework has been formulated in part I of this work for modeling reservoir compaction through linearizing coupled solid and flow field equations and constructing a local material integrator. In part II of this work, we focus on verification and performance analysis of our numerical formulation and computer implementation using several numerical examples. First, we design a cube problem in triaxial compression to verify our numerical formulation and computer code implementation especially for rock formation in compaction using cap plasticity models. The finite element prediction on stresses is compared with the analytical solution. The second problem we select is a strip footing problem popular in the geotechnical area where the evolution of soil consolidation degrees following the diffusion of pore pressure is the main interest. In this example, we demonstrate a good performance of the proposed numerical formulation on solving different shear and compaction-dominated deformation behaviors by varying the footing length. Importantly, an extremely sharp cap model based on real experimental data for Leda clays, a challenging cap model, is successfully applied in this footing problem. Our focus in this work is to model field reservoirs undergoing serious compaction. A reservoir with complex payzone geometries, multiple horizontal wells, and cap plasticity models with sharp cap surfaces has been successfully solved using our fully implicit formulation. The last example is to model a horizontal wellbore damage problem. Finally, the sensitivity of predicted subsidence to nonlinear flow model, cap hardening parameters, and Lode angles have been systemically investigated and documented in detail, which can provide a constructive guidance on how to successfully model field reservoir compaction problems with cap plasticity models.     

Simulation of non‐planar three‐dimensional hydraulic fracture propagation

Hydraulic fracturing is the method of choice to enhance reservoir permeability and well efficiency for extraction of shale gas. Multi‐stranded non‐planar hydraulic fractures are often observed in stimulation sites. Non‐planar fractures propagating from wellbores inclined from the direction of maximum horizontal stress have also been reported. The pressure required to propagate non‐planar fractures is in general higher than in the case of planar fractures. Current computational methods for the simulation of hydraulic fractures generally assume single, symmetric, and planar crack geometries. In order to better understand hydraulic fracturing in complex‐layered naturally fractured reservoirs, fully 3D models need to be developed. In this paper, we present simulations of 3D non‐planar fracture propagation using an adaptive generalized FEM. This method greatly facilitates the discretization of complex 3D fractures, as finite element faces are not required to fit the crack surfaces. A solution strategy for fully automatic propagation of arbitrary 3D cracks is presented. The fracture surface on which pressure is applied is also automatically updated at each step. An efficient technique to numerically integrate boundary conditions on crack surfaces is also proposed and implemented. Strongly graded localized refinement and analytical asymptotic expansions are used as enrichment functions in the neighborhood of fracture fronts to increase the computational accuracy and efficiency of the method. Stress intensity factors with pressure on crack faces are extracted using the contour integral method. Various non‐planar crack geometries are investigated to demonstrate the robustness and flexibility of the proposed simulation methodology. Copyright © 2014 John Wiley & Sons, Ltd.     

A Workflow for Static Reservoir Modeling Guided by Seismic Data in a Fluvial System

Realistic and accurate static geologic models are an essential element needed to predict the behavior of subsurface reservoirs and play an important role in petroleum engineering. Data used in the development of a static geologic model are gathered from various sources, such as seismic, log, and core data, each of them providing information on different physical properties of interest and with varying degrees of resolution. Compiling all data from various sources into a single representation of the subsurface formation of interest is a daily challenge for many petroleum geologists and engineers. This paper describes a framework to develop and select process-mimicking models that are consistent with available seismic attributes, namely impedance. Using a process-mimicking modeling package, 75 models of a fluvial meandering system are generated, one of which is chosen as the “true” model and masked thereafter. The implemented selection method relies on the degree of similarity in the histogram of representations of clusters of all possible patterns in the seismic impedance domain based on each process-mimicking model and that of the “true” model at several resolutions. The results demonstrate the effectiveness of the use of a weighted average divergence distance across multiple levels to select process-mimicking models that honor seismic data the best.     

正交最小二乘法及其在分析化学中的应用

介绍了一种线性模型参数回归分析方法-正交最小二乘法，并以电子探针微区分析技术分析环境样品的数据为例，对正交最小二乘法和经典最小二乘法的结果进行了详细比较。数据处理结果表明，当变自量和因变量都同时存在测量误差时（或自变量的测量误差与因变量的测量误差相比不能忽略时），正交最小二乘法获得的回归系数优于经典最小二乘法。对正交最小二乘法中的线性模型能解释的方差与经典最小二乘法中的相关系数的关系也进行了讨论。     

A shear‐dilation‐based model for evaluation of hydraulically stimulated naturally fractured reservoirs

The role of shear dilation as a mechanism of enhancing fluid flow permeability in naturally fractured reservoirs was mainly recognized in the context of hot dry rock (HDR) geothermal reservoir stimulation. Simplified models based on shear slippage only were developed and their applications to evaluate HDR geothermal reservoir stimulation were reported. Research attention is recently focused to adjust this stimulation mechanism for naturally fractured oil and gas reservoirs which reserve vast resources worldwide. This paper develops the overall framework and basic formulations of this stimulation model for oil and gas reservoirs. Major computational modules include: natural fracture simulation, response analysis of stimulated fractures, average permeability estimation for the stimulated reservoir and prediction of an average flow direction. Natural fractures are simulated stochastically by implementing ‘fractal dimension’ concept. Natural fracture propagation and shear displacements are formulated by following computationally efficient approximate approaches interrelating in situ stresses, natural fracture parameters and stimulation pressure developed by fluid injection inside fractures. The average permeability of the stimulated reservoir is formulated as a function of discretized gridblock permeabilities by applying cubic law of fluid flow. The average reservoir elongation, or the flow direction, is expressed as a function of reservoir aspect ratio induced by directional permeability contributions. The natural fracture simulation module is verified by comparing its results with observed microseismic clouds in actual naturally fractured reservoirs. Permeability enhancement and reservoir growth are characterized with respect to stimulation pressure, in situ stresses and natural fracture density applying the model to two example reservoirs. Copyright © 2002 John Wiley & Sons, Ltd.     

在储层建模中利用多点地质统计学整合地质概念模型及其解释(英文)

基于三维空间中稀疏的观测数据,地质学家和储层建模人员尝试预测井间的地质沉积相的空间非均质性时,地质概念模型和先验认识在其中扮演着重要的角色。这种整合先验模型或解释的过程有时是隐蔽或不易察觉的,正如在手工绘等值线图中的情形;它也能够被显式地运用到某种算法当中,比如数字绘图中的算法。新近兴起的多点地质统计学为地质学家和储层建模人员提供了一种有力工具,它强调使用训练图像把先验模型明确而定量地引入到储层建模当中。先验地质模型包含了被研究的真实储层中确信存在的样式,而训练图像则是该模型的定量化表达。通过再现高阶统计量,多点算法能够从训练图像中捕捉复杂的(非线性)特征样式并把它们锚定到观测的井位数据。文中描述了多点地质统计学原理,以突出训练图像概念重要性为主线,描述了多点地质统计学在建立三维储层模型中的应用。     

Application of a 3D photorealistic model for the geological analysis of the Permian carbonates (Khuff Formation) in Saudi Arabia

Three dimensional (3D) photorealistic models of geological outcrops have the potential to enhance the teaching of earth sciences by providing scale models in a virtual reality environment. These models can be run on low-cost desktop computers. Photorealistic models for geological outcrops are a digital illustration of outcrop photographs with either a point cloud representation or Triangular Irregular Network (TIN) mesh of the outcrop surface. The level of detail for these models is dependent on the target resolutions (physical and optical) that were used during data acquisition. In addition, the technique in which the data is rendered as a digital model affects the level of detail that can be observed by the geologists. A colored point cloud representation is suitable for large-scale features, but fine details are lost when the geologist zooms in to view the model close up. In contrast, a photorealistic model that is constructed from photographs draped onto a triangle mesh surface derived from Light Detection and Ranging (LiDAR) point clouds provides a level of detail that is restricted only by the resolution of the photographs.     

海洋电磁法勘探油气的三维正演模拟

针对海底可控源电磁法( MCSEM) 在油气资源勘探中的实用性,应用矢量有限元进行了多种频率的水平电偶源发射电磁场识别油气层的实例研究,并利用矢量有限元对海底的3D 模型进行了正演数值模拟。利用MCSEM 能较好地识别油气层; 归一化后的电场曲线特征表明,浅海下不易测得油气层,在实验频率内,电场源频率越高对油气层识别能力越强。同时证实了MCSEM 识别油气层是建立在波导理论基础上的。     

An assessment of the creep behaviour of Brazilian salt rocks using the multi-mechanism deformation model

Salt rocks are geomaterials that exhibit several peculiarities, which require a particular approach in rock mechanics. In the field, those rocks are usually found in layered/bedded deposits and in domes or similar structures. Creep is one of the main deformation mechanisms associated with salt rocks, and this phenomenon is highly dependent on the stress state, temperature and mineralogy. Salt rock mechanics for engineering applications requires the definition of a powerful constitutive model and this is an ongoing challenge. Among the many available models, one of the most sophisticated physical constitutive models for salt rocks is the multi-mechanism deformation creep model (MD model). The main contribution of this work is to present a first effort in the use of the MD model for Brazilian salt rocks. Material-sensitive parameters have been calibrated for the Brazilian halite through two methodologies. Salt is modelled as an elasto-viscoplastic material. Numerical simulations using the finite element method have been carried out for triaxial creep tests, Pre-salt wellbore closure and mining gallery convergence in order to validate the parameter set and the methodologies. Excellent results have been observed in most of the applications for validation. Even so, validation efforts should continue to consolidate the parameters and identify possible limitations.     

鄂尔多斯盆地海子塌地区长6油层组测井曲线标准化研究

研究区延长组沉积为陆相三角洲沉积,主要含油气层位为长6油层组.受沉积环境及泥岩中有机质含量的影响,该区非渗透层段测井曲线响应差异较大,使得测井曲线标准化效果不佳.针对此情况,在借助自然伽马、井径测井、声波测井、感应测井等曲线构建"视标准层"的基础上,利用ΔlgR方法剔除受有机质影响明显的非渗透层段,进而选取不受井眼条件、岩性、物性、含油气性等因素影响的非渗透层段作为"视标准层",最后采用直方图法对研究区测井曲线进行标准化,并结合孔隙度计算模型进行了检验.研究结果表明,ΔlgR方法可以较为有效地消除非渗透层段有机质含量的影响,提高陆相地层中"视标准层"构建的质量,为后期储层参数的确定奠定基础.     

地震储层表征:在模拟储层的三维地震响应时考虑盖层、地震勘探布局和储层物性的敏感度(英文)

根据三维地震地质模型对地震数据进行模拟是从勘探到生产的周期内决策过程中的一个不可或缺的组成部分。虽然对于在储层内的动力过程和地震地质的模型表述已经取得很大进展,但如何从这些模型得到地震数据的精确模拟仍面临很多挑战。通常是在地球模型范围内根据物性用一维褶积方法来模拟地震数据。然而这个过程一般不考虑地震勘探布局和盖层对地震信号的影响。我们审视了为什么这些因素会制约三维地球模型的有效性,并考虑了为什么需要把盖层和地震勘探布局对三维覆盖和分辨率的影响加进模拟过程之中。我们提出了一种新方法,把建立物性模型和一种新的地震模拟技术结合起来,给出一个工作流程;利用这个流程,勘探工作者可以很快模拟出三维的PSDM数据,这些数据加进了盖层和地震勘探布局对覆盖及分辨率的影响。我们利用从远离挪威海岸的一个油田得到的数据,在考虑覆盖和分辨率效应的地震数据模拟之前,对岩石物性做了一些扰动,然后进行地震数据模拟,以此来说明如何可以用这种方法提高三维地球模型的精确性和增进我们对储层的了解。     

非参数回归法在孔隙度参数预测中的应用

目前有很多利用地震属性进行储层参数预测的方法,多数方法都是针对一种特定的参数模型而制定的统计方法,即只有在一定的参数模型中才能使用的统计方法.实际应用中,由于地震属性与储层参数之间的关系十分复杂,事先无法给出合适的、具体的参数模型,使用参数模型就有可能产生较大误差.本文简要介绍了非参数回归预测方法的基本原理和方法特点,对如何分析和选取具有明确物理意义的、反映储层参数横向变化较为敏感的、与储层参数关系较为密切的地震属性进行了讨论.用非参数回归法对大港唐家河工区进行了孔隙度参数的平面分布预测,表明该方法在储层参数预测技术以及油藏描述中将会有广阔的应用前景.