Numerical Homogenization of Two-Phase Flow in Porous Media

Homogenization has proved its effectiveness as a method of upscaling for linear problems, as they occur in single-phase porous media flow for arbitrary heterogeneous rocks. Here we extend the classical homogenization approach to nonlinear problems by considering incompressible, immiscible two-phase porous media flow. The extensions have been based on the principle of preservation of form, stating that the mathematical form of the fine-scale equations should be preserved as much as possible on the coarse scale. This principle leads to the required extensions, while making the physics underlying homogenization transparent. The method is process-independent in a way that coarse-scale results obtained for a particular reservoir can be used in any simulation, irrespective of the scenario that is simulated. Homogenization is based on steady-state flow equations with periodic boundary conditions for the capillary pressure. The resulting equations are solved numerically by two complementary finite element methods. This makes it possible to assess a posteriori error bounds.     

Modelling Two-Phase Incompressible Flow in Porous Media Using Mixed Hybrid and Discontinuous Finite Elements

In this paper, we present a numerical model for simulating two-phase (oil–water and air–water) incompressible and immiscible flow in porous media. The mathematical model which is based on a fractional flow formulation is formed of two nonlinear partial differential equations: a mean pressure equation and a water saturation equation. These two equations can be solved in a sequential manner. Two numerical methods are used to discretize the equations of the two-phase flow model: mixed hybrid finite elements are used to treat the pressure equation, h-based Richards' equation and the diffusion term in the saturation equation, the advection term in the saturation equation is treated with the discontinuous finite elements. We propose a better way to calculate the nonlinear coefficients contained in our equations on each element of the discretized domain. In heterogeneous porous media, the saturation becomes discontinuous at the interface between two porous media. We show in this paper how to use the capillary pressure–saturation relationship in order to handle the saturation jump in the mixed hybrid finite element method. The two-phase flow simulator is verified against analytical solutions for some flow problems treated by other authors.     

Residual a Posteriori Error Estimates for the Mixed Finite Element Method

In this paper, residual-based a posteriori error bounds are derived for the mixed finite element method applied to a model second order elliptic problem. A global upper bound for the error in the scalar variable is established, as well as a local lower bound. In addition, due to the fact that the scalar and vector variables are approximated to equal order accuracy, the dual problem may be modified to give an upper bound for the vector variable. Some comments on estimating more general error quantities are also made. The estimate effectively guides adaptive refinement for a smooth problem with a boundary layer, as well as detects the need to refine near a singularity.     

不连续面在双重介质热-水-力三维耦合分析中的有限元数值实现

在建立双重介质热-水-力耦合微分控制方程的基础上,提出了裂隙岩体热-水-力耦合的三维力学模型,对不同介质分别建立以节点位移、水压力和温度为求解量的三维有限元格式,开发了双重介质热-水-力耦合分析的的三维有限元计算程序,在有限元数值分析中不连续面应力计算采用等厚度空间8节点节理单元进行离散,而不连续面渗流和热能计算时采用平面4节点等参单元进行离散,这样保证了不同介质之间的水量、热量交换和两类模型接触处节点水头、温度和位移相等。通过高温岩体地热开发算例,揭示了在热-水-力耦合作用下不连续面处于低应力区,其张开度随运行时间的延长呈非线性增加,非稳定渗流阶段不连续面显著地控制着渗流场的整体分布,它的水头远高于拟连续岩体介质的水头,而进入稳定渗流阶段不连续面的控渗作用不明显,由于高温岩体地热开发系统中存在大规模的热量补给,不连续面对岩体温度场分布的影响并不显著。     

A mixed finite element method for the solution of the magnetostatic problem with highly discontinuous coefficients in 3D

This paper present a new a mixed finite element method for the simulation of magnetostatic problems with highly discontinuous permeability. The method is derived from the well studied mixed formulation for the div-grad system that is known to be accurate for very large discontinuities. The method robustness is demonstrated on a test model problem.     

The Andra Couplex 1 Test Case: Comparisons Between Finite-Element, Mixed Hybrid Finite Element and Finite Volume Element Discretizations

We present here results for the Andra Couplex 1 test case, obtained with the code Cast3m. This code is developped at the CEA (Commissariat l'nergie atomique) and is used mainly to solve problems of solid mechanics, fluid mechanics and heat transfers. Different types of discretization are available, among them finite element, finite volume and mixed hybrid finite element method. Cast3m is also a componant of the platteform Alliances (co-developped by Andra, CEA), which will be used by Andra for the safety calculation of an underground waste disposal in year 2004. We solve the Darcy equation for the water flow and a convection–diffusion transport equation for the Iodine 129 which escapes from a repository cave into the water. The water flow is calculated with a MHFE discretization. It is shown that this method provides sharp results even on relatively coarse grids. The convection–diffusion transport equation is discretized with FE (Finite Element), MHFE (Mixed Hybrid Finite Element) and FV (Finite Volume) methods. In our comparison, we point out the differences of these methods in term of accuracy, respect of the maximum principle and calculations cost. Neither the finite element nor the mixed hybrid finite element approach respects the maximum principle. This results in the presence of negative concentrations near the repository cave, whereas FV calculations respect the monotonicity. We show that mass lumping techniques suppress this problem but with strong restrictions on the grid. FE and MHFE approaches are more accurate than FV for the diffusion equation, but the overall results are equivalent since the advective terms are dominant in the far field and are discretized with centered schemes. We conclude by studying the influence of the grid: a very fine grid near the repository solves almost all the problems of monotonicity, without employing mass lumping techniques. We also observed a very important increase of the accuracy on a structured grid made up of rectangles.     

基于扩展有限元的压剪复合断裂数值分析

针对岩石类材料压剪破坏,国内外学者做了大量真实岩石和模型材料的试验,并进行相应的数值模拟工作。但由于试验结果往往具有很强的离散性,且对影响试验结果的各因素缺乏系统地理论分析,加之传统常规有限元在处理裂纹这类强不连续问题时存在很大的困难,如在裂尖附近的高应力区需要剖分令人难以接受的密度的网格,同时在模拟裂纹生长时还需要对网格进行重新剖分,操作繁琐复杂,效率极低,因此,针对压剪复合断裂翼裂纹扩展规律的研究始终没有取得令人满意的成果。对原生裂纹的几何形态,主裂纹面与其间填充物的相互作用对翼裂纹起裂角的影响做了系统的理论分析,并利用扩展有限元对翼型裂纹扩展全过程进行数值模拟,与理论分析结果对比,吻合良好。相关成果为以后针对压剪复合断裂的研究提供了理论分析框架,同时对今后相关试验的开展具有理论指导作用     

Finite element analysis of the effect of corrosion on the behavior of reinforced earth walls

A simple model for the corrosion‐induced loss of stiffness and strength of the steel strips of earth‐reinforced walls was introduced in a finite element simulation of the long‐term behavior of the wall, in which the backfill‐strips interactions are taken into account by means of a generalized homogenization procedure (called a multiphase model). The results show an initial phase of slow displacements induced by the loss of stiffness, followed after a few decades by a steep acceleration of the displacements, leading to wall failure. The influences of the parameter controlling corrosion, the backfill cohesion and the heterogeneity of the corrosion process are discussed. Results are used to discuss a strategy for reinforced earth wall surveillance. Copyright © 2011 John Wiley & Sons, Ltd.     

二维接触问题的互补算法及工程应用

接触问题是一种常见的非线性问题,如何能够很好地模拟接触面的变形及受力特性,以及实现对变形体间的接触问题的真实模拟是该领域研究的难点问题。基于二维接触问题的实际物理意义,分别在法向和切向建立等价的互补模型。用非线性互补函数（NCP）中的Fischer-Burmeister（FB）函数将互补函数模型转化为非光滑方程组表达,用常规的Newton法求解。同时,基于高斯积分法可以用较少的积分点达到较高的精度,为了进一步提高求解精度,改善不连续的通病,对面-面接触模型在高斯点上对接触面进行处理,可通过调节积分点数目对求解精度进行控制,方法易于理解,实现方便。在此基础上建立二维接触有限元模型,通过一系列工程算例验证该方法的可行性与有效性。结果表明,与ABAQUS有限元的计算结果相比,该方法有着较高的精度,更真实地反映问题的实际。     

非稳定温度场的三维p型有限元方法研究

非稳定温度场的边界往往是温度梯度较大的区域,需要划分较密的网格。但在三维温度应力的计算中太密的网格往往使得计算成本很高。将p型阶谱有限元方法引入温度场的计算,则可以开拓一条解决这类问题的有效途径。研究了p型有限元中的温度边界条件和初值,编制了有关程序,对同一个算例,采用不同网格密度的常规有限元法和p型有限元法进行了温度场计算分析和对比。研究结果表明,用p型有限元进行非稳定温度场计算,在网格稀疏的条件下能够达到较高的精度。     

Numerical Homogenization of the Rigidity Tensor in Hooke''s Law Using the Node-Based Finite Element Method

Combining a geological model with a geomechanical model, it generally turns out that the geomechanical model is built from units that are at least a 100 times larger in volume than the units of the geological model. To counter this mismatch in scales, the geological data model's heterogeneous fine-scale Young's moduli and Poisson's ratios have to be upscaled to one equivalent homogeneous coarse-scale rigidity. This coarse-scale rigidity relates the volume-averaged displacement, strain, stress, and energy to each other, in such a way that the equilibrium equation, Hooke's law, and the energy equation preserve their fine-scale form on the coarse scale. Under the simplifying assumption of spatial periodicity of the heterogeneous fine-scale rigidity, homogenization theory can be applied. However, even then the spatial variability is generally so complex that exact solutions cannot be found. Therefore, numerical approximation methods have to be applied. Here the node-based finite element method for the displacement as primary variable has been used. Three numerical examples showing the upper bound character of this finite element method are presented.     

线源直流电法有限元二维正演模拟

直流电法勘探中,三维点源正演计算复杂、计算量大,虽进行傅里叶变换可转化为二维点源,但相比二维线源计算还是复杂。基于Matlab编程,采用第一类边界条件,给出了线源直流电法二维有限元数值解法,通过与均匀半空间理论值比较,验证了计算结果的正确性;再建立高阻、低阻地电模型,并采用单边供电和双边供电分别对其响应进行比较和分析,得到了相应的视电阻率断面图。结果表明单边供电可以准确反映异常体在纵向上的位置,而双边供电更能准确反应异常体的横向位置和形态。论文研究为生产实际提供了理论依据。     

构造应力场边界载荷反演的有限元逆逼近法

构造应力场问题的边界条件、加载方式是地应力场分析的难题,由于构造运动未知和地质构造复杂,无法直接求解。边界力反演是进行构造应力场研究的重要手段。从弹性力学基本方程出发,针对油气储层地应力场反问题,结合对区域构造应力场认识,利用阻尼最小二乘法建立了应力场反演的约束模型,提出了一种用于反演构造应力场边界力的反分析方法,该方法根据部分实测点的应力值进行边界力反演,并与有限元方法和有限差分法相结合,以边界力为参量逼近目标函数。根据研究区少量地应力实测资料采用该方法进行最优化计算,使得计算应力场与实测应力达到最优拟合,以弥补目前的回归反演方法和边界载荷调整法的不足来提高计算精度。算例表明了该方法的有效性,可用于解决构造应力场边界力的反演问题。     

复杂条件下3D电磁场有限元计算方法

从电磁场的Maxwell方程出发, 根据电磁场的边值问题及变分公式建立了有限元方程组.采用可以模拟较为复杂的空间地质构造和地形起伏的四面体单元离散计算区域; 单元中的插值函数选择了精度较高的十点双二次多项式; 并采用连续的双二次多项式插值函数来模拟计算区域内单元中电导率σ的空间变化.推导出了地下变电导率σ条件下计算三维电磁场的有限元单元方程的解析表达式; 采用伽辽金方法推导出了散度效正有限元方程组.根据所推导的公式, 编制了三维有限元的计算程序.数值计算结果表明, 上述公式推导正确, 为三维电磁场的数值计算提供了一条有效的新途径.      

三维锚杆数值模拟方法研究

在现有几种锚杆数值分析方法的基础上,对考虑了浆体切向受力性态的三维锚杆单元理论及其应用进行了研究,编制了三维锚杆有限元计算程序,并结合例子对浆体锚杆的变形、应力规律进行了分析。计算结果显示了模型的可靠性和实用性。     

关于构造带变形能的有限元法数学模拟

本文是在同一外力和边界及岩石力学性质条件下,用有限元数学模拟的方法研究不同构造变形带内部变形能的差别和分布问题。形状改变能（或称歪变能）ＵＷ比体积改变能（或称体变能）ＵＴ大将近一个数量级,ＵＷ、ＵＴ和构造变形能ＵＢ在挤压、剪切和引张变形岩带中总体依次逐步变小。ＵＷ和ＵＢ大致呈较大→大→小的变化,而ＵＴ却呈大→较大→小的趋势性变化。利用构造变形能这种规律,结合构造附加静水压力研究,进而探讨构造变形热及其影响流体运移等问题。     

构造应力场三维数值模拟的有限单元法

目前构造应力场的研究主要是基于平面的二维分析,随着油气成藏过程三维模拟的发展,应力场的三维数值模拟显得越来越重要.从三维空间角度出发,分析了应力场的有限元模拟与求解的整个过程.基于不规则六面体单元的角点网格数据模型,对有限元求解法进行了分析与研究并提出计算方法.基于不规则角点网格数据模型的三维构造应力场有限元计算方法是本文的创新点.最后用上述方法对东营凹陷的牛庄-王家岗区域三维应力场进行了实际模拟计算,计算结果相比于其他数据模型更符合实际情况.      

Robust numerical implementation of a 3D rate-dependent model for reservoir geomechanical simulations

A 3D elasto-plastic rate-dependent model for rock mechanics is formulated and implemented into a Finite Element (FE) numerical code. The model is based on the approach proposed by Vermeer and Neher (A soft soil model that accounts for creep. In: Proceedings of the International Symposium “Beyond 2000 in Computational Geotechnics,” pages 249-261, 1999). An original strain-driven algorithm with an Inexact Newton iterative scheme is used to compute the state variables for a given strain increment.The model is validated against laboratory measurements, checked on a simplified test case, and used to simulate land subsidence due to groundwater and hydrocarbon production. The numerical results prove computationally effective and robust, thus allowing for the use of the model on real complex geological settings.     

On the well‐posedness of some mechanical variational problems

The paper aims to establish the existence and uniqueness of the solution of some variational problems constituting the basis of finite element modellings encountered in mechanics and civil engineering. And indeed, by expanding to the approximate problems coming from the space discretization, such theoretical results contribute to strengthen the robustness of the modelling softwares and the quality of their numerical results. More particularly, three kinds of mixed variational problems involving rheological non‐linearities are considered here: the evolution problems of incompressible continua (solids or fluids) subjected to quasistatic small transformations, the problems of hydromechanical coupling and those coming from quasistatic large transformations of continua. Copyright © 2005 John Wiley & Sons, Ltd.     

三维N-S方程混合有限分析多重网格方法

提出了一种求解三维定常不可压N-S方程的混合有限分析多重网格方法(HFA-FMG).以三维方腔流动作为算例,计算表明提出的数值格式能极大地节省计算工作量且稳定有效.三维方腔流动的模拟成果较好地表现方腔内复杂的流动特征并和其他成果吻合.