A fully coupled element‐free Galerkin model for hydro‐mechanical analysis of advancement of fluid‐driven fractures in porous media

Hydraulic fracturing (HF) of underground formations has widely been used in different fields of engineering. Despite the technological advances in techniques of in situ HF, the industry uses semi‐analytical tools to design HF treatment. This is due to the complex interaction among various mechanisms involved in this process, so that for thorough simulations of HF operations a fully coupled numerical model is required. In this study, using element‐free Galerkin (EFG) mesh‐less method, a new formulation for numerical modeling of hydraulic fracture propagation in porous media is developed. This numerical approach, which is based on the simultaneous solution of equilibrium and continuity equations, considers the hydro‐mechanical coupling between the crack and its surrounding porous medium. Therefore, the developed EFG model is capable of simulating fluid leak‐off and fluid lag phenomena. To create the discrete equation system, the Galerkin technique is applied, and the essential boundary conditions are imposed via penalty method. Then, the resultant constrained integral equations are discretized in space using EFG shape functions. For temporal discretization, a fully implicit scheme is employed. The final set of algebraic equations that forms a non‐linear equation system is solved using the direct iterative procedure. Modeling of cracks is performed on the basis of linear elastic fracture mechanics, and for this purpose, the so‐called diffraction method is employed. For verification of the model, a number of problems are solved. According to the obtained results, the developed EFG computer program can successfully be applied for simulating the complex process of hydraulic fracture propagation in porous media. Copyright © 2016 John Wiley & Sons, Ltd.     

Isogeometric analysis for unsaturated flow problems

Unsaturated flow problems in porous media often described by Richards’ equation are of great importance in many engineering applications. In this contribution, we propose a new numerical flow approach based on isogeometric analysis (IGA) for modeling the unsaturated flow problems. The non-uniform rational B-spline (NURBS) basis is utilized for spatial discretization whereas the stable implicit backward Euler method for time discretization. The nonlinear Richards’ equation is iteratively solved with the aid of the Newton–Raphson scheme. Owing to some desirable features of an efficient numerical flow approach, major advantages of the present formulation involve: (a) numerical oscillation at the wetting front can be avoided or facilitated, simply by using either an h-refinement or a lumped mass matrix technique; (b) higher-order exactness can be obtained due to the nature of the IGA features; (c) the approach is straightforward to implement and it does not need any transformation, e.g., Kirchhoff transformation or filter algorithm; and (d) in contrast to the Picard iteration scheme, which forms linear convergences, the proposed approach can however yield quadratic convergences by using the Newton–Raphson method for solving resultant nonlinear equations. Numerical model validation is analyzed by solving a three-dimensional unsaturated flow problem in soil, and its derived results are verified against analytical solutions. Numerical applications are then studied by considering three extensive examples with simple and complex configurations to further show the accuracy and applicability of the present IGA.     

山东淄博市大武水源地裂隙岩溶水中污染物运移的数值研究

在分析研究淄博市大武水源地裂隙岩溶含水层的水力性质和污染物运移特征的基础上 ,对裂隙岩溶水的水头和污染物运移进行数值研究。目前国内外对裂隙岩溶水进行数值计算时 ,通常用等价多孔介质模型 ,但裂隙岩溶介质和多孔介质有很大不同。裂隙岩溶介质的储水和导水空间为裂隙网络 ,导水系数大 ,地下水的实际平均流速比孔隙水大得多 ,但给水度和贮水系数小。当用等价多孔介质模型进行模拟时应考虑这些特点。对于污染物运移的模拟 ,要同时求解水头方程和对流弥散方程 ,可采用MODFLOW和MT3D软件进行模拟。研究区裂隙岩溶水水头的数值计算表明 ,等效多孔介质模型水头的拟合误差能满足国标GB/T144 97- 93的要求。各时段地下水水量均衡计算的精度也满足要求。对流弥散方程的数值计算 ,由于Peclet数高达 95 .6 7,对流占绝对优势 ,可能存在数值弥散和数值振荡 ,因而采用多种方法进行了比较。对于同一问题 ,同时采用上游有限差分法 (UFDM) ,混合的欧拉拉格朗日方法 (特征线法MOC、改进特征线法MMOC和混合特征线法HMOC) ,总变异消减法(TVD)进行计算 ,并比较其结果。结果表明 ,混合特征线法 (HMOC)和总变异消减法 (TVD)比较适合于对流占优势的运移问题计算。由于渗透系数K和有效孔隙度θ对溶质运移结果的影响很大 ,?     

孔隙网络模型在土壤水文学中的应用研究进展

非饱和土壤的水力性质是采用模型定量模拟水分和溶质在非饱和带中运动的最重要的物理参数,可以用网络模型来预测。孔隙网络模型的主要优点在于可以对发生在土壤孔隙尺度上的物理、化学过程进行直观的表达和模拟。目前,国内外研究者已在样品图像获取、图像分析、模型建立以及求解等方面取得了一定的进展。在对其进行综合评述的基础上,指出了已有网络模型在建立过程中存在的不足之处,并进一步明确了今后的研究方向。     

Numerical simulation of ground anchors

A procedure of finite-element modeling and beam-column modeling of ground anchors was proposed in this study to investigate the load transfer mechanism in ground anchors. The procedure included the modeling of soil, grout, and strand tendon and the interface modeling of soil–grout and grout–strand in ground anchors. A series of finite element analyses and beam-column analyses were performed using the proposed models on ground anchors. The numerical predictions were compared with observed measurements in a field load test. The results indicated that the numerical simulation of load transfer mechanism on ground anchors can provide reasonable predictions.     

沙质河床冲刷过程中床沙级配的模拟

研究了沙质河床冲刷过程中床沙级配的模拟方法。在该方法中采用了新的描述床沙级配变化的动力学方程和床沙级配在交换层内垂向变化与水流强度、河床冲刷强度的关系,反映了冲刷过程中交换层内向床面补充的物理过程。通过室内实验率定了相关参数,建立了床沙级配变化动力学方程的数值求解方法,模拟了床沙冲刷粗化。计算结果与室内实验结果符合较好,表明所建议的方法反映了河床冲刷中床沙级配变化的物理事实,可用于沙质河床冲刷粗化模拟。     

离心实验在污染物迁移研究中的应用

离心实验模拟以其可以获得与原型一致的应力水平并且能够大大缩短原型历时而被应用于污染物迁移实验研究。在离心相似理论及离心实验模拟的相似基础上,重点回顾了离心机在饱和带水分、非饱和带水分、保守性溶质、NAPLs、重金属、核素迁移以及污染场地修复方面的实验应用。最后讨论了离心模拟中土壤预制和加速度选择的问题,并简单介绍了离心监测方法。可以得出结论:离心机能够成功用于各类物质迁移的实验研究中,离心实验模拟能够为理论和数值等分析方法快速提供真实可靠的参数依据,但离心实验的理论基础和监测方法需要进一步完善,此外也应积极开展更接近实际情况下的离心实验模拟研究。     

边坡稳定分析的极限分析下限解有限元法

研制开发了极限分析下限解有限元程序.在程序的开发过程中,着重解决了下限解有限元法转化为标准线性规划问题和线性优化方法的选择问题,同时,通过典型算例的分析对比,对程序的合理性和可行性进行了验证.     

Comparison of numerical methods for simulating strongly nonlinear and heterogeneous reactive transport problems—the MoMaS benchmark case

Although multicomponent reactive transport modeling is gaining wider application in various geoscience fields, it continues to present significant mathematical and computational challenges. There is a need to solve and compare the solutions to complex benchmark problems, using a variety of codes, because such intercomparisons can reveal promising numerical solution approaches and increase confidence in the application of reactive transport codes. In this contribution, the results and performance of five current reactive transport codes are compared for the 1D and 2D subproblems of the so-called easy test case of the MoMaS benchmark (Carrayrou et al., Comput Geosci, 2009, this issue). This benchmark presents a simple fictitious reactive transport problem that highlights the main numerical difficulties encountered in real reactive transport problems. As a group, the codes include iterative and noniterative operator splitting and global implicit solution approaches. The 1D easy advective and 1D easy diffusive scenarios were solved using all codes, and, in general, there was a good agreement, with solution discrepancies limited to regions with rapid concentration changes. Computational demands were typically consistent with what was expected for the various solution approaches. The differences between solutions given by the three codes solving the 2D problem are more important. The very high computing effort required by the 2D problem illustrates the importance of parallel computations. The most important outcome of the benchmark exercise is that all codes are able to generate comparable results for problems of significant complexity and computational difficulty.     

实现复频移完全匹配层吸收边界条件的递推卷积方法

完全匹配层吸收边界（PML）已经被证明是非常有效的边界吸收技术,对体波和面波的吸收都具有非常好的效果,已经被广泛应用于弹性波的数值模拟中。但是在某些情况下传统的PML技术还是存在一定的问题,比如对掠射情况下的体波和窄区域自由表面条件下的面波的吸收等等。在坐标变换中采用复频移拉伸函数的复频移PML可以有效地改善PML边界条件的吸收性能。基于弹性波一阶速度-应力方程,推导了复频移PML的递推卷积实现方法,并采用交错网格高阶有限差分法对其进行了数值模拟,与传统的PML进行了对比。结果表明：传统的PML对掠射情况下的体波和窄区域自由表面条件下的面波吸收不足,会产生虚假反射,影响真实波场;而基于递推卷积的复频移PML算法能够有效地改善困难情况下的吸收效果,并且在实现过程中不用分裂变量,应用更加方便简单。计算卷积时采用递推的形式,推导过程直观易懂,易于编程,而且不会增加计算量,存储量也没有太大的变化。     

Conservative, shock-capturing transport methods with nonconservative velocity approximations

Conservative high-resolution, or shock-capturing, methods have become widely used for modeling transport equations described by conservation laws. In many geoscience applications, the transport equation is coupled to a conservation or continuity equation for a velocity field. Depending on how the velocity is approximated, the continuity equation may or may not be satisfied, either locally or globally. In this paper, we discuss the effect this has on a typical high resolution scheme, and propose a correction which accounts for the fact that the velocity may be nonconservative. We present several numerical examples and prove stability bounds and an a priori error estimate for the corrected method.     

A supercoarsening multigrid method for poroelasticity in 3D coupled flow and geomechanics modeling

The Galerkin finite-element discretization of the force balance equation typically leads to large linear systems for geomechanical problems with realistic dimensions. In iteratively coupled flow and geomechanics modeling, a large linear system is solved at every timestep often multiple times during coupling iterations. The iterative solution of the linear system stemming from the poroelasticity equations constitutes the most time-consuming and memory-intensive component of coupled modeling. Block Jacobi, LSOR, and Incomplete LU factorization are popular preconditioning techniques used for accelerating the iterative solution of the poroelasticity linear systems. However, the need for more effective, efficient, and robust iterative solution techniques still remains especially for large coupled modeling problems requiring the solution of the poroelasticity system for a large number of timesteps. We developed a supercoarsening multigrid method (SCMG) which can be multiplicatively combined with commonly used preconditioning techniques. SCMG has been tested on a variety of coupled flow and geomechanics problems involving single-phase depletion and multiphase displacement of in-situ hydrocarbons, CO₂ injection, and extreme material property contrasts. Our analysis indicates that the SCMG consistently improves the convergence properties of the linear systems arising from the poroelasticity equations, and thus, accelerates the coupled simulations for all cases subject to investigation. The joint utilization of the two-level SCMG with the ILU1 preconditioner emerges as the most optimal preconditioning/iterative solution strategy in a great majority of the problems evaluated in this work. The BiCGSTAB iterative solver converges more rapidly compared to PCG in a number of test cases, in which various SCMG-accelerated preconditioning strategies are applied to both iterators.     

A new data assimilation procedure to develop a debris flow run-out model

Parameter calibration is one of the most problematic phases of numerical modeling since the choice of parameters affects the model’s reliability as far as the physical problems being studied are concerned. In some cases, laboratory tests or physical models evaluating model parameters cannot be completed and other strategies must be adopted; numerical models reproducing debris flow propagation are one of these. Since scale problems affect the reproduction of real debris flows in the laboratory or specific tests used to determine rheological parameters, calibration is usually carried out by comparing in a subjective way only a few parameters, such as the heights of soil deposits calculated for some sections of the debris flows or the distance traveled by the debris flows using the values detected in situ after an event has occurred. Since no automatic or objective procedure has as yet been produced, this paper presents a numerical procedure based on the application of a statistical algorithm, which makes it possible to define, without ambiguities, the best parameter set. The procedure has been applied to a study case for which digital elevation models of both before and after an important event exist, implicating that a good database for applying the method was available. Its application has uncovered insights to better understand debris flows and related phenomena.     

一维畦灌施肥地表水流与溶质运移耦合模型——Ⅰ.模型建立

提高畦灌施肥地表水流与溶质运移数值模型的稳定性、收敛性及计算精度,有利于改善地面畦灌施肥系统的设计与评价工作.在利用隐-显混合时间格式对一维畦灌地表水流与溶质运移耦合模型中包含的各矢量项进行时间离散基础上,借助有限差分法、有限体积法和有限单元法分别对由隐时间格式生成的物理矢量线性近似式空间导数、物理矢量空间导数、溶质扩散矢量和地形矢量项进行空间离散,对最终形成的控制方程代数方程组进行数值求解,构建起基于混合数值解法的一维畦灌施肥地表水流与溶质运移耦合模型.     

Mortar coupling and upscaling of pore-scale models

Pore-scale models are becoming increasingly useful as predictive tools for modeling flow and transport in porous media. These models can accurately represent the 3D pore-structure of real media. Currently first-principles modeling methods are being employed for obtaining qualitative and quantitative behavior. Generally, artificial, simple boundary conditions are imposed on a model that is used as a stand-alone tool for extracting macroscopic parameters. However, realistic boundary conditions, reflecting flow and transport in surrounding media, may be necessary for behavior that occurs over larger length scales or including pore-scale models in a multiscale setting. Here, pore-scale network models are coupled to adjacent media (additional pore-scale or continuum-scale models) using mortars. Mortars are 2D finite-element spaces employed to couple independent subdomains by enforcing continuity of pressure and flux at shared boundary interfaces. While mortars have been used in the past to couple subdomains of different models, physics, and meshes, they are extended here for the first time to pore-scale models. The approach is demonstrated by modeling single-phase flow in coupled pore-scale models, but the methodology can be utilized to model dynamic processes and perform multiscale modeling in 3D continuum simulators for flow and transport.     

Finite element methods for variable density flow and solute transport

Saltwater intrusion into coastal freshwater aquifers is an ongoing problem that will continue to impact coastal freshwater resources as coastal populations increase. To effectively model saltwater intrusion, the impacts of increased salt content on fluid density must be accounted for to properly model saltwater/freshwater transition zones and sharp interfaces. We present a model for variable density fluid flow and solute transport where a conforming finite element method discretization with a locally conservative velocity post-processing method is used for the flow model and the transport equation is discretized using a variational multiscale stabilized conforming finite element method. This formulation provides a consistent velocity and performs well even in advection-dominated problems that can occur in saltwater intrusion modeling. The physical model is presented as well as the formulation of the numerical model and solution methods. The model is tested against several 2-D and 3-D numerical and experimental benchmark problems, and the results are presented to verify the code.     

长江中下游江湖水沙调控数值模拟

以长江中下游防洪系统为对象,在大型复杂防洪系统洪水运动数值模拟基础上,成功地将面向长江中下游防洪规划论证需求的洪水演进数学模型转化为面向长江防洪系统江湖水沙调控需求的长江中下游江湖水沙调控数学模型.为适应江湖水沙调控和评估的要求,提出了基于水动力学的闸坝调度计算模式.此外,还对河网分汊泥沙分配模式进行了深入研究.通过长江中下游防洪规划及其洞庭湖区"控支强干"方案论证模拟计算,较好地解决了防洪措施蓄泄后效评估和工程优化调控模式等关键难题,为防洪规划方案的制定提供了定量依据,主要成果已应用于长江中下游防洪规划方案.     

Numerical modeling of the flow and transport of radionuclides in heterogeneous porous media

This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation and a conservative finite volume scheme for the concentration equation. The convective term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion–dispersion term is discretized by piecewise linear conforming triangular finite elements. It is shown that the scheme satisfies a discrete maximum principle. Numerical examples demonstrate the effectiveness of the methodology for a coupled system that includes an elliptic equation and a diffusion–convection–reaction equation arising when modeling flow and transport in heterogeneous porous media. The proposed scheme is robust, conservative, efficient, and stable, as confirmed by numerical simulations.      

山东烟台夹河中、下游地区海水入侵三维水质数值模拟研究

建立了三维变密度对流弥散水质数学模型来研究山东省烟台夹河中、下游地区咸淡水界面的运移规律。以四面体为基本离散单元 ,推导出三维海水入侵变密度水质模型求解的数值方法 ,其中水流方程求解时运用了迦辽金有限单元法。溶质运移方程求解时运用了欧拉拉格朗日混合方法 ,将对流项与弥散项分离 ,用传统迦辽金有限元方法求解弥散项 ;采用自适应MOC MMOC法求解对流项 ,以消除人工过量和数值弥散。根据地下水的潮汐效应观测信息 ,确定了含水系统的海底延伸边界 ;利用该地区地下水水头及水质长观资料识别了模型的水文地质参数 ,探讨了夹河地区海水入侵的原因 :认为夹河下游地区滨海地带地下水过量开采是造成烟台地区海水入侵的主要原因。此外 ,海水随潮定期地倒灌进入夹河 ,通过局部岩性天窗侵入淡水含水层加剧了沿夹河河床两侧地下水的咸化。同时还预测了几种情况下地下水的水质演化趋势 ,为防止和减轻夹河地区海水入侵提供合理、科学的依据。