异质地层稳定渗流场随机多尺度有限元模拟研究

摘要

This paper studies on the stable seepage field of heterogeneous strata in small scale by stochastic multiscale finite element method. The method is different from traditional stochastic finite elements for such as problems. The mulitiscale propriety of strata is not considered by traditional methods, so those methods cannot resolve the multiscale propriety of nature strata in a trivial means. So a new method called stochastic heterogeneous multiscale finite element method (SHMFE) is introduced to resolve the problem efficiently. The In permeability Y=lnKε in micro-scale is described by Karhunen-Lopteve expansion. In order to make the problem well-defined, the stochastic collocation method is adopted by combining generalized polynomial chaos to disperse the probability space. If Karhunen-Lobteve expansion has a lot of random variable, sparse grid stochastic collocation method is used. The well-defined problem is solved by stochastic multiscale finite element method. The numerical example argues that stochastic multiscale finite element method is better than traditional methods in the aspects of computational resource and cost. It can resolve the nature of multiscale heterogeneous porous media which classical methods do not.%为了模拟异质多尺度地层稳定渗流场，提出了一种结合有限元异质多尺度法（FEHM）与随机配点法（SCM）的随机异质多尺度有限元法（SHMFE）。目前，研究此类问题的主要手段是随机有限元法（SFE），但是这类方法无法从本质上表达地层属性的异质多尺度问题。当地层属性如渗透系数的异质性所在的尺度远远小于研究区域时，运用传统的方法模拟其异质性需大量计算资源，以现有物质基础是一个不可完成的任务。SHMFE能有效地解决这类多尺度不均匀问题。首先运用Karhunen-Loμeve（KL）分解在细尺度上将对数渗透系数Y=lnKε进行展开，然后采用耦合广义多项式混沌的SCM法离散问题概率空间使之成为确定性问题，如果KL展开有多个随机变量，则采用稀疏随机配点法；最后采用FEHM法求解此确定性问题。计算实例表明，相对于传统的SFE，SHMFE能利用更少的计算资源有效地模拟本质上多尺度的渗流问题。