Ordering-based nonlinear solver for fully-implicit simulation of three-phase flow

The Fully Implicit method (FIM) is often the method of choice for the temporal discretization of the partial differential equations governing multiphase flow in porous media. The FIM involves solving large coupled systems of nonlinear algebraic equations. Newton-based methods, which are employed to solve the nonlinear systems, can suffer from convergence problems—this is especially true for large time steps in the presence of highly nonlinear flow physics. To overcome such convergence problems, the time step is usually reduced, and the Newton steps are restarted from the solution of the previous (converged) time step. Recently, potential ordering and the reduced-Newton method were used to solve immiscible three-phase flow in the presence of buoyancy and capillary effects (e.g., Kwok and Tchelepi, J. Comput. Phys. 227(1), 706–727 2007). Here, we improve the robustness of the potential-based ordering method in the presence of gravity. Furthermore, we also extend this nonlinear approach to interphase mass transfer. Our algorithm deals effectively with mass transfer between the liquid and gas phases, including phase disappearance (e.g., gas going back in solution) and reappearance (e.g., gas coming out of solution and forming a separate phase), as a function of pressure and composition. Detailed comparisons of the robustness and efficiency of the potential-based solver with state-of-the-art nonlinear/linear solvers are presented for immiscible two-phase (Dead-Oil), Black-Oil, and compositional problems using heterogeneous models. The results show that for large time steps, our nonlinear ordering-based solver reduces the number of nonlinear iterations significantly, which leads to gains in the overall computational cost.     

Ordering-based nonlinear solver for fully-implicit simulation of three-phase flow

The Fully Implicit Method (FIM) is often the method of choice for the temporal discretization of the partial differential equations governing multiphase flow in porous media. The FIM involves solving large coupled systems of nonlinear algebraic equations. Newton-based methods, which are employed to solve the nonlinear systems, can suffer from convergence problems—this is especially true for large time steps in the presence of highly nonlinear flow physics. To overcome such convergence problems, the time step is usually reduced, and the Newton steps are restarted from the solution of the previous (converged) time step. Recently, potential ordering and the reduced-Newton method were used to solve immiscible three-phase flow in the presence of buoyancy and capillary effects (e.g., Kwok and Tchelepi, J. Comput. Phys. 227(1), 706–727 9). Here, we improve the robustness of the potential-based ordering method in the presence of gravity. Furthermore, we also extend this nonlinear approach to interphase mass transfer. Our algorithm deals effectively with mass transfer between the liquid and gas phases, including phase disappearance (e.g., gas going back in solution) and reappearance (e.g., gas coming out of solution and forming a separate phase), as a function of pressure and composition. Detailed comparisons of the robustness and efficiency of the potential-based solver with state-of-the-art nonlinear/linear solvers are presented for immiscible two-phase (Dead-Oil), Black-Oil, and compositional problems using heterogeneous models. The results show that for large time steps, our nonlinear ordering-based solver reduces the number of nonlinear iterations significantly, which leads to gains in the overall computational cost.     

A parallel global-implicit 2-D solver for reactive transport problems in porous media based on a reduction scheme and its application to the MoMaS benchmark problem

In this article, an approach for the efficient numerical solution of multi-species reactive transport problems in porous media is described. The objective of this approach is to reformulate the given system of partial and ordinary differential equations (PDEs, ODEs) and algebraic equations (AEs), describing local equilibrium, in such a way that the couplings and nonlinearities are concentrated in a rather small number of equations, leading to the decoupling of some linear partial differential equations from the nonlinear system. Thus, the system is handled in the spirit of a global implicit approach (one step method) avoiding operator splitting techniques, solved by Newton’s method as the basic algorithmic ingredient. The reduction of the problem size helps to limit the large computational costs of numerical simulations of such problems. If the model contains equilibrium precipitation-dissolution reactions of minerals, then these are considered as complementarity conditions and rewritten as semismooth equations, and the whole nonlinear system is solved by the semismooth Newton method.     

Swarm intelligence-based solver for parameter estimation of laboratory through-diffusion transport of contaminants

Theoretical approaches are of fundamental importance to predict the potential impact of waste disposal facilities on ground water contamination. Appropriate design parameters are generally estimated by fitting theoretical models to data gathered from field monitoring or laboratory experiments. Transient through-diffusion tests are generally conducted in the laboratory to estimate the mass transport parameters of the proposed barrier material. These parameters are usually estimated either by approximate eye-fitting calibration or by combining the solution of the direct problem with any available gradient-based techniques. In this work, an automated, gradient-free solver is developed to estimate the mass transport parameters of a transient through-diffusion model. The proposed inverse model uses a particle swarm optimization (PSO) algorithm that is based on the social behavior of animals searching for food sources. The finite difference numerical solution of the forward model is integrated with the PSO algorithm to solve the inverse problem of parameter estimation. The working principle of the new solver is demonstrated and mass transport parameters are estimated from laboratory through-diffusion experimental data. An inverse model based on the standard gradient-based technique is formulated to compare with the proposed solver. A detailed comparative study is carried out between conventional methods and the proposed solver. The present automated technique is found to be very efficient and robust. The mass transport parameters are obtained with great precision.     

推移质输沙率的非线性研究

目的是从非线性科学角度探讨泥沙运动的规律。通过分析非均匀沙起动的影响因素得到了尖点突变模型的状态变量和控制变量,建立了能描述推移质运动的尖点突变模型。在尖点突变标准方程的基础上,应用尖点突变理论的坐标变换和拓扑变换,导出了输沙强度和水流参数的函数关系式。将输沙强度作为状态变量,水流参数和床沙密实系数作为控制变量。用水槽实验资料和其它推移质输沙率公式进行了对比验证。验证结果表明,计算值与其它推移质输沙率公式和水槽实验结果基本相符,误差一般在-90%～80%之间。说明建立的推移质输沙率公式是合理的,能够反映泥沙的起动和输移规律。     

Fast linear solver for diffusion problems with applications to pressure computation in layered domains

Accurate simulation of fluid pressures in layered reservoirs with strong permeability contrasts is a challenging problem. For this purpose, the Discontinuous Galerkin (DG) method has become increasingly popular. Unfortunately, standard linear solvers are usually too inefficient for the aforementioned application. To increase the efficiency of the conjugate gradient (CG) method for linear systems resulting from symmetric interior penalty (discontinuous) Galerkin (SIPG) discretizations, we cast an existing two-level preconditioner into the deflation framework. The main idea is to use coarse corrections based on the DG solution with polynomial degree p = 0. This paper provides a numerical comparison of the performance of the original preconditioner and the resulting deflation variant in terms of scalability and overall efficiency. Furthermore, it studies the influence of the SIPG penalty parameter, weighted averages in the SIPG formulation (SWIP), the smoother, damping of the smoother, and the strategy for solving the coarse systems. We have found that the penalty parameter can best be chosen diffusion-dependent. In that case, both two-level methods yield fast and scalable convergence. Whether preconditioning or deflation is to be favored depends on the choice of the smoother and on the damping of the smoother. Altogether, both two-level methods can contribute to cheaper and more accurate fluid pressure simulations.     

基于精确Riemann求解器的明满流过渡过程模拟

Preissmann窄缝法模拟明满流过渡过程方法简单,但存在明显的非物理振荡,抑制非物理振荡是该方法应用的关键。基于Godunov格式和精确Riemann求解器对明满流过渡过程进行模拟,针对Riemann问题代数恒等式在明满流交界处不光滑问题,提出了三阶收敛方法与二分法结合的迭代求解方法,保证迭代收敛至真实解;针对由于变量空间重构方法不能准确表达变量在空间中真实物理状态而导致的非物理振荡,提出了基于精确Riemann解的变量空间重构方法,准确表达激波间断在单元内的空间分布状态,从机理上抑制了非物理振荡。实例研究表明,数值计算结果与解析解或实测值吻合良好,研究成果为明满流过渡过程的高精度数值模拟提供了新的方法。     

基于精确Riemann求解器的明满流过渡过程模拟

Preissmann窄缝法模拟明满流过渡过程方法简单,但存在明显的非物理振荡,抑制非物理振荡是该方法应用的关键。基于Godunov格式和精确Riemann求解器对明满流过渡过程进行模拟,针对Riemann问题代数恒等式在明满流交界处不光滑问题,提出了三阶收敛方法与二分法结合的迭代求解方法,保证迭代收敛至真实解;针对由于变量空间重构方法不能准确表达变量在空间中真实物理状态而导致的非物理振荡,提出了基于精确Riemann解的变量空间重构方法,准确表达激波间断在单元内的空间分布状态,从机理上抑制了非物理振荡。实例研究表明,数值计算结果与解析解或实测值吻合良好,研究成果为明满流过渡过程的高精度数值模拟提供了新的方法。     

A tomographic imagery segmentation methodology for three-phase geomaterials based on simultaneous region growing

X-ray computed tomography is a powerful non-destructive technique used in many domains to obtain the three-dimensional representation of objects, starting from the reconstitution of two-dimensional images of radiographic scanning. This technique is now able to analyze objects within a few micron resolutions. Consequently, X-ray microcomputed tomography opens perspectives for the analysis of the fabric of multiphase geomaterials such as soils, concretes, rocks and ceramics. To be able to characterize the spatial distribution of the different phases in such complex and disordered materials, automated phase recognition has to be implemented through image segmentation. A crucial difficulty in segmenting images lies in the presence of noise in the obtained tomographic representation, making it difficult to assign a specific phase to each voxel of the image. In the present study, simultaneous region growing is used to reconstitute the three-dimensional segmented image of granular materials. First, based on a set of expected phases in the image, regions where specific phases are sure to be present are identified, leaving uncertain regions of the image unidentified. Subsequently, the identified regions are grown until growing phases meet each other with vanishing unidentified regions. The method requires a limited number of manual parameters that are easily determined. The developed method is illustrated based on three applications on granular materials, comparing the phase volume fractions obtained by segmentation with macroscopic data. It is demonstrated that the algorithm rapidly converges and fills the image after a few iterations.     

A reactive transport simulator for variable porosity problems

ReactMiCP, a new reactive transport simulator was developed based on the semismooth speciation solver SpecMiCP. Its main feature is a sequential iterative operator splitting algorithm where macroscopic model parameters are explicitly included in the formulation. Its correctness, robustness, and efficiency are tested against the MoMaS benchmark and two sets of cement paste lab experiments. We show that a robust speciation solver is a key requirement for good performance of the reactive transport simulator. We also demonstrate that a sequential iterative solver should be preferred over non-iterative solvers when using operator splitting. The flexibility and the speed of the simulator are used to test the influence of the database, the initial condition, and the diffusion coefficient model for the cement paste simulations.     

Theoretical connections between optimization algorithms based on an approximate gradient

Performing a line search method in the direction given by the simplex gradient is a well-known method in the mathematical optimization community. For reservoir engineering optimization problems, both a modification of the simultaneous perturbation stochastic approximation (SPSA) and ensemble-based optimization (EnOpt) have recently been applied for estimating optimal well controls in the production optimization step of closed-loop reservoir management. The modified SPSA algorithm has also been applied to assisted history-matching problems. A recent comparison of the performance of EnOpt and a SPSA-type algorithm (G-SPSA) for a set of production optimization test problems showed that the two algorithms resulted in similar estimates of the optimal net-present-value and required roughly the same amount of computational time to achieve these estimates. Here, we show that, theoretically, this result is not surprising. In fact, we show that both the simplex, preconditioned simplex, and EnOpt algorithms can be derived directly from a modified SPSA-type algorithm where the preconditioned simplex algorithm is presented for the first time in this paper. We also show that the expectation of all these preconditioned stochastic gradients is a first-order approximation of the preconditioning covariance matrix times the true gradient or a covariance matrix squared times the true gradient.     

基于分形先验信息的非线性反演

基于分形先验信息的非线性反演方法能综合利用测井数据和地震信息,在贝叶斯框架下,通过分形高斯噪音算法得到基于分形理论的先验信息,然后根据地震资料构建似然函数,最终利用基于快速模拟退火算法(Very Fast Simulated Annealing,VFSA)改进的粒子群优化(Particle Swarm Optimization,PSO)算法(VFSA-PSO)实现后验概率密度的抽样。与确定性反演结果相比,该方法能够有效地融合测井资料中的高频信息,提高反演结果的分辨率,并且目标函数的建立融合了确定性反演中的低频约束,从而得到宽频带的反演结果。数值模拟试验表明:基于分形先验的非线性反演结果与理论模型吻合较好,实际资料的应用效果也证明了该反演方法的有效性。     

基于主成分分析的重力侵蚀区域提取方法研究

历史重力侵蚀是研究区域重力侵蚀危险性的一个重要数据,通过多期航片、卫星影像结合野外调查是快速确定重力侵蚀发生区域的主要方法.以马来西亚金马仑高原作为研究区,选取不同时相的SPOT5数据,运用主成分分析方法,提取发生重力侵蚀的区域.研究表明,通过多时相主成分分析得到的第二主成分主要反映了地表植被的变化信息,运用阈值法可快速提取可能发生重力侵蚀的区域.     

饱和水流溶质运移问题数值解法综述

本文总结了饱和水流中溶质运移方程求解的各种数值方法，分析各种方法的本质特征以及各自的优缺点，并指出了求解对流—弥散方程的各种数值方法的研究进展和值得重视的问题。研究结果表明，自适应欧拉—拉格朗日法(EM)是溶质运移问题中，求解对流—弥散方程是比较有发展潜力的方法之一。以MMOC法为基础在陡峰值高价插值和其它区域低价插值相结合的ELM法，将是未来发展的趋势。而寻求非规则网格上高精度的空间单元插值模式，已开始成为求解对流问题数值方法研究的重点和关键问题。     

非线性回归分析的PSO小波网络方法及应用

文章介绍了粒子群算法（PSO）和小波神经网络的基本原理，把基于粒子群小波网络的混合算法应用到非线性回归问题中，并对算法解决非线性回归问题进行了实践分析，最后建立了测井响应值和物性参数孔隙度之间的回归模型。从仿真结果可以看出，本方法的回归值和岩心分析值符合较好，表明粒子群小波网络进行非线性回归分析是一种有效的数据回归方法。     

Mixed multiscale finite element methods using approximate global information based on partial upscaling

The use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information. The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the fine-scale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308–317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods.     

利用算子分裂迎风均衡格式解对流为主溶质运移问题

水污染模拟问题是水流问题与溶质运移问题的耦合问题.各种常见的数值解法在以对流为主溶质运移问题的求解中都会遇到困难,如用有限单元法或有限差分法时,会产生数值弥散与过量这两类误差.引入算子分裂迎风均衡格式法求解对流为主的水污染模拟问题,较好地克服了数值弥散和数值解出现振荡问题,该格式具有良好的稳定性、单调性及守恒性特点.     

A GCV based method for nonlinear ill-posed problems

This paper discusses the inversion of nonlinear ill-posed problems. Such problems are solved through regularization and iteration and a major computational problem arises because the regularization parameter is not known a priori. In this paper we show that the regularization should be made up of two parts. A global regularization parameter is required to deal with the measurement noise, and a local regularization is needed to deal with the nonlinearity. We suggest the generalized cross validation (GCV) as a method to estimate the global regularization parameter and the damped Gauss-Newton to impose local regularization. Our algorithm is tested on the magnetotelluric problem. In the second part of this paper we develop a methodology to implement our algorithm on large-scale problems. We show that hybrid regularization methods can successfully estimate the global regularization parameter. Our algorithm is tested on a large gravimetric problem. This revised version was published online in July 2006 with corrections to the Cover Date.     

Confidence regions in ternary diagrams based on the power-divergence statistics

Watson and Nguyen (1985) and Watson (1987) consider the problem of plotting confidence regions in a ternary diagram, for a trinomial probability vector , based on Pearson's x². Their results are extended to the power-divergence family of statistics resulting in confidence regions of diverse shapes and sizes. The members of the family with the most accurate coverage probabilities are =2/3 and =1/2.