基于多尺度有限元法的频率域航空电磁三维正演研究

正演模拟是电磁数据反演的基础, 其计算速度与精度一直是制约电磁反演的两大核心问题.在三维电磁正反演中, 传统方法通过加密网格或增加插值基函数阶数提高计算精度, 但由此也降低了计算效率, 制约了三维电磁反演的实用化.因此, 如何实现大尺度模型高精度快速正演是目前电磁三维正反演中亟需解决的问题.本文将多尺度有限元法应用到麦克斯韦方程求解中.我们首先在粗网格尺度上构建满足局部特性微分算子的多尺度基函数, 进而在粗网格尺度上对原问题进行求解, 通过建立粗细两套网格间场的映射关系, 在未知数较少的粗网格上实现电磁问题求解之后, 利用粗细两套网格间场的映射关系获取细网格上电磁场响应, 由此可以在保证计算精度前提下快速获取不同尺度电磁场正演响应, 计算速度得到很大提高.此外, 本文还基于八叉树思想进行网格优化, 进一步改善三维正演效率.我们通过对典型地电结构进行多尺度有限元正演模拟并与传统有限元结果对比验证算法的有效性.最后, 我们通过模拟加拿大Voisey's Bay卵形体镍铜硫化矿区航空电磁响应以检验本文算法模拟地下复杂异常体的能力.

     

Coupling multiscale finite element method for consolidation analysis of heterogeneous saturated porous media

The multiscale finite element method is developed for solving the coupling problems of consolidation of heterogeneous saturated porous media under external loading conditions. Two sets of multiscale base functions are constructed, respectively, for the pressure field of fluid flow and the displacement field of solid skeleton. The coupling problems are then solved with a multiscale numerical procedure in space and time domain. The heterogeneities induced by permeabilities and mechanical parameters of the saturated porous media are both taken into account. Numerical experiments are carried out for different cases in comparison with the standard finite element method. The numerical results show that the coupling multiscale finite element method can be successfully used for solving the complicated coupling problems. It reduces greatly the computing effort in both memory and time for transient problems.     

Coarsening of three-dimensional structured and unstructured grids for subsurface flow

We present a generic, semi-automated algorithm for generating non-uniform coarse grids for modeling subsurface flow. The method is applicable to arbitrary grids and does not impose smoothness constraints on the coarse grid. One therefore avoids conventional smoothing procedures that are commonly used to ensure that the grids obtained with standard coarsening procedures are not too rough. The coarsening algorithm is very simple and essentially involves only two parameters that specify the level of coarsening. Consequently the algorithm allows the user to specify the simulation grid dynamically to fit available computer resources, and, e.g., use the original geomodel as input for flow simulations. This is of great importance since coarse grid-generation is normally the most time-consuming part of an upscaling phase, and therefore the main obstacle that has prevented simulation workflows with user-defined resolution. We apply the coarsening algorithm to a series of two-phase flow problems on both structured (Cartesian) and unstructured grids. The numerical results demonstrate that one consistently obtains significantly more accurate results using the proposed non-uniform coarsening strategy than with corresponding uniform coarse grids with roughly the same number of cells.     

Density estimation of two-phase flow with multiscale and randomly perturbed data

In this paper, we describe an efficient approach for quantifying uncertainty in two-phase flow applications due to perturbations of the permeability in a multiscale heterogeneous porous medium. The method is based on the application of the multiscale finite element method within the framework of Monte Carlo simulation and an efficient preprocessing construction of the multiscale basis functions. The quantities of interest for our applications are the Darcy velocity and breakthrough time and we quantify their uncertainty by constructing the respective cumulative distribution functions. For the Darcy velocity we use the multiscale finite element method, but due to lack of conservation, we apply the multiscale finite volume element method as an alternative for use with the two-phase flow problem. We provide a number of numerical examples to illustrate the performance of the method.     

A note on variational multiscale methods for high-contrast heterogeneous porous media flows with rough source terms

In this short note, we discuss variational multiscale methods for solving porous media flows in high-contrast heterogeneous media with rough source terms. Our objective is to separate, as much as possible, subgrid effects induced by the media properties from those due to heterogeneous source terms. For this reason, enriched coarse spaces designed for high-contrast multiscale problems are used to represent the effects of heterogeneities of the media. Furthermore, rough source terms are captured via auxiliary correction equations that appear in the formulation of variational multiscale methods [23]. These auxiliary equations are localized and one can use additive or multiplicative constructions for the subgrid corrections as discussed in the current paper. Our preliminary numerical results show that one can capture the effects due to both spatial heterogeneities in the coefficients (such as permeability field) and source terms (e.g., due to singular well terms) in one iteration. We test the cases for both smooth source terms and rough source terms and show that with the multiplicative correction, the numerical approximations are more accurate compared to the additive correction.     

一种基于亥姆霍兹分解的大地电磁测深有限元正演预条件解法

本研究针对大地电磁测深法有限元数值模拟中,迭代法求解线性方程组效率较低的问题,利用亥姆霍兹分解原理,将电场矢量双旋度方程的预条件问题转化为基于矢量位的泊松问题和基于标量位的拉普拉斯问题,并在四面体非结构化棱边元离散的情况下,借助节点元辅助网格离散上述预条件问题,进一步利用代数多重网格方法（AMG）实施求解,最终实现预条件算法.利用经典的COMMEMI理论模型进行试算并与前人的积分方程解进行对比,验证了本文数值模拟程序与预条件方法的正确性和可靠性.此外,利用不同自由度规模的实验模型对这一预条件算法的效率进行了测试.结果表明,这一算法可以有效地提升大地电磁测深法棱边有限元数值模拟迭代法的收敛性,计算效率较通用的不完全LU分解预条件算法明显更高;在较大自由度网格（>1000万）数值模拟计算中,其算法效率及内存占用相对直接解法有较大优势,也使小型工作站上利用较大自由度的有限元网格进行大地电磁测深数值模拟计算成为可能.

     

Bayesian uncertainty quantification for flows in heterogeneous porous media using reversible jump Markov chain Monte Carlo methods

In this paper, we study the uncertainty quantification in inverse problems for flows in heterogeneous porous media. Reversible jump Markov chain Monte Carlo algorithms (MCMC) are used for hierarchical modeling of channelized permeability fields. Within each channel, the permeability is assumed to have a log-normal distribution. Uncertainty quantification in history matching is carried out hierarchically by constructing geologic facies boundaries as well as permeability fields within each facies using dynamic data such as production data. The search with Metropolis–Hastings algorithm results in very low acceptance rate, and consequently, the computations are CPU demanding. To speed-up the computations, we use a two-stage MCMC that utilizes upscaled models to screen the proposals. In our numerical results, we assume that the channels intersect the wells and the intersection locations are known. Our results show that the proposed algorithms are capable of capturing the channel boundaries and describe the permeability variations within the channels using dynamic production history at the wells.     

Comparison of free-surface and rigid-lid finite element models of barotropic instabilities

The main goal of this work is to appraise the finite element method in the way it represents barotropic instabilities. To that end, three different formulations are employed. The free-surface formulation solves the primitive shallow-water equations and is of predominant use for ocean modeling. The vorticity–stream function and velocity–pressure formulations resort to the rigid-lid approximation and are presented because theoretical results are based on the same approximation. The growth rates for all three formulations are compared for hyperbolic tangent and piecewise linear shear flows. Structured and unstructured meshes are utilized. The investigation is also extended to time scales that allow for instability meanders to unfold, permitting the formation of eddies. We find that all three finite element formulations accurately represent barotropic instablities. In particular, convergence of growth rates toward theoretical ones is observed in all cases. It is also shown that the use of unstructured meshes allows for decreasing the computational cost while achieving greater accuracy. Overall, we find that the finite element method for free-surface models is effective at representing barotropic instabilities when it is combined with an appropriate advection scheme and, most importantly, adapted meshes.     

A 2D finite volume model for bebris flow and its application to events occurred in the Eastern Pyrenees

FLATModel is a 2D finite volume code that contains several original approaches to improve debris-flow simulation. Firstly, FLATModel incorporates a ＂stop-and-go＂ technique in each cell to allow continuous collapses and remobilizations of the debris-flow mass. Secondly, flow velocity and consequently yield stress is directly associated with the type of rheology to improve boundary accuracy. Thirdly, a simple approach for entrainment is also included in the model to analyse the effect of basal erosion of debris flows. FLATMODEL was tested at several events that occurred in the Eastern Pyrenees and simulation results indicated that the model can represent rather well the different characteristics observed in the field.     

A multiscale method for subsurface inverse modeling: Single-phase transient flow

High-resolution geologic models that incorporate observed state data are expected to effectively enhance the reliability of reservoir performance prediction. One of the major challenges faced is how to solve the large-scale inverse modeling problem, i.e., to infer high-resolution models from the given observations of state variables that are related to the model parameters according to some known physical rules, e.g., the flow and transport partial differential equations. There are typically two difficulties, one is the high-dimensional problem and the other is the inverse problem. A multiscale inverse method is presented in this work to attack these problems with the aid of a gradient-based optimization algorithm. In this method, the model responses (i.e., the simulated state data) can be efficiently computed from the high-resolution model using the multiscale finite-volume method. The mismatch between the observations and the multiscale solutions is then used to define a proper objective function, and the fine-scale sensitivity coefficients (i.e., the derivatives of the objective function with respect to each node’s attribute) are computed by a multiscale adjoint method for subsequent optimization. The difficult high-dimensional optimization problem is reduced to a one-dimensional one using the gradient-based gradual deformation method. A synthetic single-phase transient flow example problem is employed to illustrate the proposed method. Results demonstrate that the multiscale framework presented is not only computationally efficient but also can generate geologically consistent models. By preserving spatial structure for inverse modeling, the method presented overcomes the artifacts introduced by the multiscale simulation and may enhance the prediction ability of the inverse-conditional realizations generated.     

Coupling upscaling finite element method for consolidation analysis of heterogeneous saturated porous media

The coupling upscaling finite element method is developed for solving the coupling problems of deformation and consolidation of heterogeneous saturated porous media under external loading conditions. The method couples two kinds of fully developed methodologies together, i.e., the numerical techniques developed for calculating the apparent and effective physical properties of the heterogeneous media and the upscaling techniques developed for simulating the fluid flow and mass transport properties in heterogeneous porous media. Equivalent permeability tensors and equivalent elastic modulus tensors are calculated for every coarse grid block in the coarse-scale model of the heterogeneous saturated porous media. Moreover, an oversampling technique is introduced to improve the calculation accuracy of the equivalent elastic modulus tensors. A numerical integration process is performed over the fine mesh within every coarse grid element to capture the small scale information induced by non-uniform scalar field properties such as density, compressibility, etc. Numerical experiments are carried out to examine the accuracy of the developed method. It shows that the numerical results obtained by the coupling upscaling finite element method on the coarse-scale models fit fairly well with the reference solutions obtained by traditional finite element method on the fine-scale models. Moreover, this method gets more accurate coarse-scale results than the previously developed coupling multiscale finite element method for solving this kind of coupling problems though it cannot recover the fine-scale solutions. At the same time, the method developed reduces dramatically the computing effort in both CPU time and memory for solving the transient problems, and therefore more large and computational-demanding coupling problems can be solved by computers.     

3D finite volume groundwater and heat transport modeling with non-orthogonal grids,using a coordinate transformation method

Many popular groundwater modeling codes are based on the finite differences or finite volume method for orthogonal grids. In cases of complex subsurface geometries this type of grid either leads to coarse geometric representations or to extremely fine meshes. We use a coordinate transformation method (CTM) to circumvent this shortcoming. In computational fluid dynamics (CFD), this method has been applied successfully to the general Navier–Stokes equation. The method is based on tensor analysis and performs a transformation of a curvilinear into a rectangular unit grid, on which a modified formulation of the differential equations is applied. Therefore, it is not necessary to reformulate the code in total. We applied the CTM to an existing three-dimensional code (SHEMAT), a simulator for heat conduction and advection in porous media. The finite volume discretization scheme for the non-orthogonal, structured, hexahedral grid leads to a 19-point stencil and a correspondingly banded system matrix. The implementation is straightforward and it is possible to use some existing routines without modification. The accuracy of the modified code is demonstrated for single phase flow on a two-dimensional analytical solution for flow and heat transport. Additionally, a simple case of potential flow is shown for a two-dimensional grid which is increasingly deformed. The result reveals that the corresponding error increases only slightly. Finally, a thermal free-convection benchmark is discussed. The result shows, that the solution obtained with the new code is in good agreement with the ones obtained by other codes.     

A new multiscale scheme for computing statistical moments in single phase flow in heterogeneous porous media

In this work we develop a new multiscale procedure to compute numerically the statistical moments of the stochastic variables which govern single phase flow in heterogeneous porous media. The technique explores the properties of the log-normally distributed hydraulic conductivity, characterized by power-law or exponential covariances, which shows invariance in its statistical structure upon a simultaneous change of the scale of observation and strength of heterogeneity. We construct a family of equivalent stochastic hydrodynamic variables satisfying the same flow equations at different scales and strengths of heterogeneity or correlation lengths. Within the new procedure the governing equations are solved in a scaled geology and the numerical results are mapped onto the original medium at coarser scales by a straightforward rescaling. The new procedure is implemented numerically within the Monte Carlo algorithm and also in conjunction with the discretization of the low-order effective equations derived from perturbation analysis. Numerical results obtained by the finite element method show the accuracy of the new procedure to approximated the two first moments of the pressure and velocity along with its potential in reducing drastically the computational cost involved in the numerical modeling of both power-law and exponential covariance functions.     

Development of elasto-plastic viscous damper finite element model for reinforced concrete frames

By advancing the technologies regarding seismic control of structures and development of earthquake resistance systems in the past decades application of different types of earthquake energy dissipation system has incredibly increased. Viscous damper device as a famous and the simplest earthquake energy dissipation system is implemented in many new structures and numerous number of researches have been done on the performance of viscous dampers in structures subjected to earthquake. The experience of recent severe earthquakes indicates that sometimes the earthquake energy dissipation devices are damaged during earthquakes and there is no function for structural control system. So, damage of earthquake energy dissipation systems such as viscous damper device must be considered during design of earthquake resistance structures.This paper demonstrates the development of three-dimensional elasto-plastic viscous damper element consisting of elastic damper in the middle part and two plastic hinges at both ends of the element which are compatible with the constitutive model to reinforce concrete structures and are capable to detect failure and damage in viscous damper device connections during earthquake excitation. The finite element model consists of reinforced concrete frame element and viscous damper element is developed and special finite element algorithm using Newmark׳s direct step-by-step integration is developed for inelastic dynamic analysis of structure with supplementary elasto-plastic viscous damper element. So based on all the developed components an especial finite computer program has been codified for “Nonlinear Analysis of Reinforced Concrete Buildings with Earthquake Energy Dissipation System”. The evaluation of seismic response of structure and damage detection in structural members and damper device was carried out by 3D modeling, of 3 story reinforced concrete frame building under earthquake multi-support excitation.     

Parallel finite element algorithm with domain decomposition for three-dimensional magnetotelluric modelling

We present a new finite element (FE) method for magnetotelluric modelling of three-dimensional conductivity structures. Maxwell's equations are treated as a system of first-order partial differential equations for the secondary fields. Absorbing boundary conditions are introduced, minimizing undesired boundary effects and allowing the use of small computational domains. The numerical algorithm presented here is an iterative, domain decomposition procedure employing a nonconforming FE space. It does not use global matrices, therefore allowing the modellization of large and complicated structures. The algorithm is naturally parallellizable, and we show results obtained in the IBM SP2 parallel supercomputer at Purdue University. The accuracy of the numerical method is verified by checking the computed solutions with the results of COMMEMI, the international project on the comparison of modelling methods for electromagnetic induction.     

基于非结构网格有限元的三维井地电阻率法任意各向异性正演模拟

地层介质的电各向异性增加了井地电阻率法响应的复杂性, 开展基于电各向异性介质模型的井地电阻率法响应规律研究对于正确解释各向异性显著地区的观测数据特征至关重要.针对垂直线源井地电阻率法的任意各向异性响应模拟问题, 本文提出了一种基于非结构网格有限元三维正演算法, 通过引入3×3的对称正定张量来表征任意各向异性的电导率, 采用非结构四面体网格有限元方法来离散电位的边值问题, 通过将垂直线源等效为一系列点源问题, 进而实现了任意各向异性介质中井地电阻率法的高效数值计算.通过与三个地电模型的解的对比, 验证了本文数值解算法的精度和有效性.针对线源远离和垂直穿过异常体的两类模型, 分别考察了当围岩或异常体为各向异性介质时的井地视电阻率响应特征.结果表明, 对于各向异性地层, 围岩和异常体的主轴电阻率值和旋转角均会对井地视电阻率的幅值及分布产生显著的影响.研究结果对于提高井地电阻率法的认识和资料解释水平具有重要的理论和实际意义.

     

修正辛格式有限元法的地震波场模拟

三角网格有限元法具有网格剖分的灵活性,能有效模拟地震波在复杂介质中的传播.但传统有限元法用于地震波场模拟时计算效率较低,消耗较大计算资源.本文采用改进的核矩阵存储（IKMS）策略以提高有限元法的计算效率,该方法不用组合总体刚度矩阵,且相比于常规有限元法节省成倍的内存.对于时间离散,将有限元离散后的地震波运动方程变换至Hamilton体系,在显式二阶辛Runge-Kutta-Nystr m（RKN）格式的基础之上加入额外空间离散算子构造修正辛差分格式,通过Taylor展开式得到具有四阶时间精度时间格式,且辛系数全为正数.本文从理论上分析了时空改进方法相比传统辛-有限元方法在频散压制、稳定性提升等方面的优势.数值算例进一步证实本方法具有内存消耗少、稳定性强和数值频散弱等优点.

     

Mass-conservative finite volume methods on 2-D unstructured grids for the Richards’ equation

The solution to the 2-D time-dependent unsaturated flow equation is numerically approximated by a second-order accurate cell-centered finite-volume discretization on unstructured grids. The approximation method is based on a vertex-centered Least Squares linear reconstruction of the solution gradients at mesh edges.A Taylor series development in time of the water content dependent variable in a finite-difference framework guarantees that the proposed finite volume method is mass conservative. A Picard iterative scheme solves at each time step the resulting non-linear algebraic problem. The performance of the method is assessed on five different test cases and implementing four distinct soil constitutive relationships. The first test case deals with a column infiltration problem. It shows the capability of providing a mass-conservative behavior. The second test case verifies the numerical approximation by comparison with an analytical mixed saturated–unsaturated solution. In this case, the water drains from a fully saturated portion of a 1-D column. The third and fourth test cases illustrate the performance of the approximation scheme on sharp soil heterogeneities on 1-D and 2-D multi-layered infiltration problems. The 2-D case shows the passage of an abrupt infiltration front across a curved interface between two layers. Finally, the fifth test case compares the numerical results with an analytical solution that is developed for a 2-D heterogeneous soil with a source term representing plant roots. This last test case illustrates the formal second-order accuracy of the method in the numerical approximation of the pressure head.     

Two-dimensional finite volume method for dam-break flow simulation

A numerical model based upon a second-order upwind ceil-center f＇mite volume method on unstructured triangular grids is developed for solving shallow water equations. The assumption of a small depth downstream instead of a dry bed situation changes the wave structure and the propagation speed of the front which leads to incorrect results. The use of Harten-Lax-vau Leer （HLL） allows handling of wet/dry treatment. By usage of the HLL approximate Riemann solver, also it make possible to handle discontinuous solutions. As the assumption of a very small depth downstream oftbe dam can change the nature of the dam break flow problem which leads to incorrect results, the HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is applied to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order Runge-Kutta method is used for the time integration of semi-discrete equations. The developed numerical model has been applied to several test cases as well as to real flows. The tests are tested in two cases： oblique hydraulic jump and experimental dam break in converging-diverging flume. Numerical tests proved the robustness and accuracy of the model. The model has been applied for simulation of dam break analysis of Torogh in Iran. And finally the results have been used in preparing EAP （Emergency Action Plan）.