地下水模型的Neumann展开Monte-Carlo随机有限元法

讨论了求解地下水模型(水流模型和水质模型)的Neumann展开Monte-Carlo随机有限元法。从基本的随机变量入手,避免了过程中随机变量的增多问题,给出结点水头(浓度)的均值、方差和水头在某区间的概率计算方法;改进了矩阵求逆的效率,对输入随机变量较多、随机变量变异较大的非稳定地下水问题特别有效。同时选取二维承压地下水水流问题(有解析解)作为例子,进行了随机数值模拟实验。      

A surrogate-based simulation–optimization approach application to parameters’ identification for the HydroGeoSphere model

A systematic approach is needed to use water more productively, because water shortages limit socio-economic development in many parts of the world. The aim of this paper is to establish a surrogate-based simulation–optimization approach to identify parameter values for a fully integrated surface water and groundwater flow coupling simulation. A surface water and groundwater flow coupling simulation model was implemented using HydroGeoSphere (HGS) model and the parameter sensitivities in the model were analyzed using local sensitivity analysis method. The parameters that exerted a large influence on the output results of the HGS model were then selected as stochastic variables, and the stochastic variable data sets were generated using the latin hypercube sampling (LHS) method, which, thereby, were used as inputs in HGS model to obtain the corresponding outputs. On the basis of input and output data sets, a kriging surrogate model of the HGS model was then established and verified, and parameter values of HGS model were identified using a surrogate-based simulation–optimization approach. The results of this study show that parameters that exert a large influence on the simulation output results include hydraulic conductivity, porosity, the van genuchten parameter (\(\alpha\)), and channel manning coefficient. The established kriging surrogate model is an ideal alternative to the HGS model for simulating and predicting, while optimal parameter values can be identified effectively and accurately using the established approach. The results of this research reveal that huge computational loads can be mitigated while using the kriging surrogate as an alternative for a simulation model in the solution process of optimization model.     

地下水动态观测网优化设计研究

本文采用状态空间分析法,通过对地下水流系统输入变量、输出变量及地下水动态测量值的研究,提出了地下水流系统确定一随机性数值模型。运用该模型的模拟解,进行地下水动态观测网优化设计。该方法把地下水动态观测网密度、位置及观测频率优化与地下水流系统结合起来研究。为了克服该模型计算量大的缺陷,本文提出了一种改进算法,使这一观测网优化方法适于在微机上实现,减少了计算工作量。该方法曾用于陕西某地区地下水位动态观测网的优化设计,取得了显著经济效益。     

基于随机理论的地下水环境风险评价

针对确定性模型难以描述含水层非均质空间分布的问题,提出基于随机理论的地下水环境风险评价方法。以矩形场地地下水污染风险评价为例,采用蒙特卡罗法生成大量渗透系数随机场,模拟含水层参数各种可能的非均质空间分布,在此基础上建立场地地下水流模型与溶质运移模型,分别计算污染物在地下水中的迁移转化情况。统计大量随机模拟中污染事故发生的频率,当模拟次数足够多时,污染频率收敛于污染概率,污染风险即通过污染概率体现出来。该方法将模型参数设为满足一定分布特征的随机变量,避免了确定性方法得出的武断的评价结果,可为工厂的选址、水源地的选址等工作提供科学指导。     

玛纳斯河流域地下水水位变化及水量平衡研究

天山北坡玛纳斯河流域地下水长期大规模开采引发了一系列生态环境问题,地下水均衡状态和地下水流动模式明显发生了改变。为研究节水灌溉条件下的地下水系统流动模式及其对地下水均衡要素的改变规律,以平原绿洲区为研究对象,采用三维地下水数值模拟方法,研究玛纳斯河流域地下水水位动态变化及水量平衡规律。结果表明:研究区地下水水位的抬升和回落受农业灌溉的影响显著,具有一定的周期性,存在着明显的时空差异;地下水均衡处于负均衡状态,补排差为-2.81×108m3。模拟期内观测水头与计算水头两者相关系数各月均在0.81以上,模拟效果较好。     

挠力河流域三维地下水流数值模拟

在分析概化挠力河流域地质及水文地质条件的基础上,建立了研究区地下水流系统的三维数学模拟模型,运用Visual MODFLOW软件进行求解,并对研究区未来7年的地下水流场进行了预报。模型预报过程中用迭代逼近方法预报二类边界,用频谱分析法预报年降水量和地下水开采量,实现了随机模型与确定性模型相耦合。结果表明,研究区地下水多年平均补给量为20.9×10⁸ m³/a。从预报的等水位线图看出,按照预测的开采量进行开采,研究区的地下水位在预报期间略有下降,每年平均下降0.329 m。     

潜水摄动有限元的随机数值模拟

潜水水流的动态随机模拟是一个复杂而难解决的问题.通过建立二维潜水非稳定流模拟的摄动随机有限元模型, 把控制方程的主要参数渗透系数和给水度随机变量、及源汇项和边界条件看作随机变量.在充分考虑4种随机因素的条件下, 推导出求解潜水二维非稳定流均值和方差的9个方程; 重点介绍了不同方程数值离散的特殊处理方法.通过设定理想例子对模拟结果进行了分析, 表明随机变量中边界条件值方差、渗透系数方差变化对水头方差变化的影响很小, 给水度方差的变化对水头方差的变化影响很大.本模型考虑因素全面, 对一般的潜水非稳定流随机模拟都可应用.本研究给出了边界、渗透系数、给水度的随机因素对潜水动态模拟的影响, 丰富和补充了地下水运动的随机理论.      

白杨水源地潜水水质的Monte-Carlo随机模拟

对白杨水源地的地质和水文地质条件进行简要论述,在此基础上建立了二维潜水地下水水流和水质的数学模型。根据渭河水位、水质、降水量等随机变量的特征,选取Ｍｏｎｔｅ－Ｃａｒｌｏ随机模拟方法对水源地潜水的水流和氯离子浓度进行了评价和预测,求出了潜水水位和氯离子浓度的均值及方差,并对氯离子浓度超过１００ｍｇ／Ｌ的概率给出了定量预测,得出了水源地投产后总输水管道中水的氯离子浓度与渭河水中氯离子浓度密切相关等结论。     

含水层和土壤的随机特征对水分运动的影响

将土壤或含水层的介质参数视为随机分布,建立了饱和非饱和水流运动的随机微分方程,利用随机数值求解方法,得到各种随机参数对水头分布和含水量空间变异性的影响。分析结果表明,随着渗透系数空间变异性和相关尺度的增大,含水层水头分布的变异性增大;对于区域内有抽水井分布的问题,在井点附近区域,z向水流速度空间变异性大,而在井点处变异性较小。当采用Gardner-Russo模型描述非饱和水分特征时,其非饱和土壤介质参数α的变异性对土壤负压和含水量变异性的影响大于饱和渗透系数,土壤含水量越小,含水量的空间变异性越大。初步讨论了根据随机数值方法研究地下水运动问题的可靠性分析方法。     

利用趋势化随机参数场的地下水流数值模拟优化方法

水文地质参数场的刻画是建立地下水流数值模拟模型的关键问题和难点问题。通常来讲,参数场合理性程度越高,模型拟合精度越高。本次研究将随机方法和参数空间分布表达进行结合,提出了趋势化随机参数场的构建方法。以渗透系数为研究对象,首先利用MCMC采样和样本数据特征确定水文地质参数的基本数据结构,进而根据样本空间分布特征对其进行趋势化处理,最终形成趋势化的渗透系数场。通过算例分析,利用趋势化处理后的渗透系数场能够大幅提高模拟精度,相比传统赋均值方法其误差可降至原来的1/3。在北京大兴跌隆起地区进行的实例应用说明,趋势化渗透系数场对提升岩性粒径较大(中砂以上)地区模拟精度效果显著,案例中粗砂区域渗透系数经趋势化处理后平均拟合误差由2.76 m下降至0.64 m;而对岩性以细砂及以下粒径为主的区域模拟精度提升并不明显。总体来说,该方法可为地下水流数值模型的优化提供借鉴,提升模型拟合精度,从而更加合理地刻画地下水流系统。     

基于区域分解法的地下水有限元与边界元耦合模型—淄博市王旺庄水源地地下水数值模拟

区域分解法（DDM）是20世纪90年代兴起的一种求解偏微分方程的新方法，方法本身独到的耦合思想和高效的并行计算机理，对于求解复杂的、大型的地下水问题具有相当的优势和广阔的应用前景。本文以淄博市王旺庄水源地地下水流模型为例，应用重叠型区域分解法（DDM）构造了边界单元法（BEM）与有限单元法（FEM）耦合模型，在两种数值方法各自优点的基础上，更形象地再现了实际水文地质原型，有效地消除了人为边界造成的流场失真。     

Stochastic Modeling of Variably Saturated Transient Flow in Fractal Porous Media

This work presents the application of a Monte Carlo simulation method to perform an statistical analysis of transient variably saturated flow in an hypothetical random porous media. For each realization of the stochastic soil parameters entering as coefficients in Richards' flow equation, the pressure head and the flow field are computed using a mixed finite element procedure for the spatial discretization combined with a backward Euler and a modified Picard iteration in time. The hybridization of the mixed method provides a novel way for evaluating hydraulic conductivity on interelement boundaries. The proposed methodology can handle both large variability and fractal structure in the hydraulic parameters. The saturated conductivity K _s and the shape parameter _vg in the van Genuchten model are treated as stochastic fractal functions known as fractional Brownian motion (fBm) or fractional Gaussian noise (fGn). The statistical moments of the pressure head, water content, and flow components are obtained by averaging realizations of the fractal parameters in Monte Carlo fashion. A numerical example showing the application of the proposed methodology to characterize groundwater flow in highly heterogeneous soils is presented.     

地下水流系统理论与研究方法的发展

地下水流系统理论的提出,推动了现代水文地质学的发展。以Tóth经典地下水流系统理论建立方法为基础,综述了基于Tóth方法的地下水流系统的模拟成果,分析了Tóth方法得出的水流模式与控制因素的关系,以及地下水流系统理论从概念到实际应用的发展。同时,对中国地质大学(武汉)提出的地下水流系统通量上边界模拟方法也进行了系统论述,在物理模拟实验与数值模拟基础上,认为通量上边界分析方法是对Tóth方法的改进与完善,有利于对地下水流系统发育的物理机制理解;该方法能够更全面认识地下水流系统模式及其转化,定量出各影响因素对地下水流模式的控制关系。最后指出地下水流系统理论是当代水文地质学的核心概念框架,应该重视地下水流系统理论物理机制和数学模拟方法的研究,加强新技术方法的引入,拓宽其应用领域的研究等。     

Groundwater head uncertainty analysis in unsteady-state water flow models using the interval and perturbation methods

In the numerical simulation of groundwater flow, uncertainties often affect the precision of the simulation results. Stochastic and statistical approaches such as the Monte Carlo method, the Neumann expansion method and the Taylor series expansion, are commonly employed to estimate uncertainty in the final output. Based on the first-order interval perturbation method, a combination of the interval and perturbation methods is proposed as a viable alternative and compared to the well-known equal interval continuous sampling method (EICSM). The approach was realized using the GFModel (an unsaturated-saturated groundwater flow simulation model) program. This study exemplifies scenarios of three distinct interval parameters, namely, the hydraulic conductivities of six equal parts of the aquifer, their boundary head conditions, and several hydrogeological parameters (e.g. specific storativity and extraction rate of wells). The results show that the relative errors of deviation of the groundwater head extremums (RDGE) in the late stage of simulation are controlled within approximately ±5% when the changing rate of the hydrogeological parameter is no more than 0.2. From the viewpoint of the groundwater head extremums, the relative errors can be controlled within ±1.5%. The relative errors of the groundwater head variation are within approximately ±5% when the changing rate is no more than 0.2. The proposed method of this study is applicable to unsteady-state confined water flow systems.

     

基于三维随机水文地质结构模型的地下水流数值模拟: 以西辽河平原为例

为了探究水文地质结构对地下水流数值模拟的不确定性,可以运用随机模拟建立地下水位的预测模型。根据转移概率地质统计方法模拟多孔介质岩性分布,利用非线性规划的思路计算岩性与水文地质参数之间的关系,从而建立相对精确的随机水文地质参数场。将不同的水文地质参数场运用到MODFLOW中,得到不同的随机模拟结果。通过比较随机模型和确定模型的末流场拟合情况以及水位动态拟合图,发现确定模型和随机模型具有相似趋势,都能与实测流场拟合较好,但是随机模型更能体现真实的水文地质特征。对随机模型预测10年后的地下水水位做不确定性分析,得到水位平均变幅介于-5～5 m之间,且95%置信度水位变幅的平均上限线约为0.146 m。研究结果为决策者提供科学依据。     

地下水位随机性影响下的边坡可靠性分析

在边坡稳定性评价模型中可将地下水的水压力总分为静水压力和动水压力。建立考虑水压力的剩余推力法边坡稳定性力学分析模型。推导了利用已知两点地下水位求解其间任一点地下水位和含水层厚度的求解公式。将地下水位和滑面的抗剪强度指标作为随机变量,利用已有数据求其分布特征,提出基于Monte-Carlo法与剩余推力法的边坡可靠性分析方法,并对某公路边坡进行可靠性分析。计算结果表明地下水位的变化与边坡失效概率的变化成正相关关系。在地下水位变化较大的7、8、9月,将地下水位作为随机变量计算得到的边坡失效概率为60%左右,而将地下水位作为确定量计算的边坡失效概率为10%~30%;在地下水位变化不大的其它月份,二者差别不大,但前者总大于后者。     

盆地地下水流系统形成与影响因素分析

目前区域（盆地）地下水流系统模拟研究中,常用的定水头与通量两种上边界条件刻画方法与实际条件存在差距。通过对比两种方法的差异和各自适用条件,采用解析法讨论地下水位的形成控制机制,提出了改进后的变通量上边界数值模型,并以鄂尔多斯盆地北部白垩系地下水流系统为例分析了盆地地下水流系统的形成与影响因素。研究表明,鄂尔多斯盆地北部白垩系水流系统地下水位受地形、补给条件和渗透系数三者共同控制,同时特有的气候、地形和岩性组合通过控制地下水位影响地下水流系统的发育演化。采用变通量上边界法探讨上边界条件改变对盆地水流系统的影响,对深刻认识区域地下水流系统形成演化机制,揭示地下水系统与上边界气候变化、植被生态变化之间的相互作用关系具有一定优势。     

含水层层状非均质对地下水流系统的影响

区域尺度上含水层非均质具有复杂的结构性和随机性,难以准确刻画,造成非均质对区域地下水流系统的影响机制研究不够深入。本文以鄂尔多斯盆地白垩系地下水流系统为研究实例,选择典型剖面,采用剖面二维随机数值模拟方法,通过对比不同非均质刻画方法下地下水流场的变化,探讨含水层层状非均质对地下水流系统的影响机制。结果显示,均质条件下模型各向异性（含水层水平和垂向渗透系数比值Kh/Kv）取值为1000时,地下水流场与实际条件较为接近;非均质条件下,渗透系数方差取值0.91,水平相关长度取值5000 m,Kh/Kv取值150时,接近实际条件。研究表明,在大尺度地下水流模拟研究中,采用水平相关长度、渗透系数方差和各向异性值三个变量生成的随机场能很好地刻画含水层的层状非均质特征及其对水流系统的影响控制作用。由于含水层不同尺度层状非均质的叠加效应,采用均质各向异性介质等效概化含水层层状非均质性会造成等效各向异性值偏大失真的效应。     

Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables

This paper proposes a stochastic response surface method for reliability analysis involving correlated non-normal random variables, in which the Nataf transformation is adopted to effectively transform the correlated non-normal variables into independent standard normal variables. Transformations of random variables that are often used in reliability analyses in terms of standard normal variables are summarized. The closed-form expressions for fourth to sixth order Hermite polynomial chaos expansions involving any number of random variables are formulated. The proposed method will substantially extend the application of stochastic response surface method for reliability problems. An example of reliability analysis of rock slope stability with plane failure is presented to demonstrate the validity and capability of the proposed stochastic response surface method. The results indicate that the proposed stochastic response surface method can evaluate the reliability of rock slope stability involving correlated non-normal variables accurately and efficiently. Its accuracy is shown to be higher than that for the first-order reliability method, and it is much more efficient than direct Monte-Carlo simulation. The results also show that the number of collocation points selected should ensure that the Hermite polynomial matrix has a full rank so that different order SRSMs can produce a robust estimation of probability of failure for a specified performance function. Generally, the accuracy of SRSM increases as the order of SRSM increases.     

An integrative approach to robust design and probabilistic risk assessment for CO<Subscript>2</Subscript> storage in geological formations

CO₂ storage in geological formations is currently being discussed intensively as a technology with a high potential for mitigating CO₂ emissions. However, any large-scale application requires a thorough analysis of the potential risks. Current numerical simulation models are too expensive for probabilistic risk analysis or stochastic approaches based on a brute-force approach of repeated simulation. Even single deterministic simulations may require parallel high-performance computing. The multiphase flow processes involved are too non-linear for quasi-linear error propagation and other simplified stochastic tools. As an alternative approach, we propose a massive stochastic model reduction based on the probabilistic collocation method. The model response is projected onto a higher-order orthogonal basis of polynomials to approximate dependence on uncertain parameters (porosity, permeability, etc.) and design parameters (injection rate, depth, etc.). This allows for a non-linear propagation of model uncertainty affecting the predicted risk, ensures fast computation, and provides a powerful tool for combining design variables and uncertain variables into one approach based on an integrative response surface. Thus, the design task of finding optimal injection regimes explicitly includes uncertainty, which leads to robust designs with a minimum failure probability. We validate our proposed stochastic approach by Monte Carlo simulation using a common 3D benchmark problem (Class et al., Comput Geosci 13:451–467, 2009). A reasonable compromise between computational efforts and precision was reached already with second-order polynomials. In our case study, the proposed approach yields a significant computational speed-up by a factor of 100 compared with the Monte Carlo evaluation. We demonstrate that, due to the non-linearity of the flow and transport processes during CO₂ injection, including uncertainty in the analysis leads to a systematic and significant shift of the predicted leakage rates toward higher values compared with deterministic simulations, affecting both risk estimates and the design of injection scenarios.