多尺度有限单元法求解非均质多孔介质中的三维地下水流问题

doi:10.11867/j.issn.1001-8166.2004.03.0437

地球科学进展 ›› 2004, Vol. 19 ›› Issue (3): 437 -442. doi: 10.11867/j.issn.1001-8166.2004.03.0437

叶淑君;吴吉春;薛禹群

南京大学地球科学系，江苏南京210093

收稿日期:2004-04-09 修回日期:2004-04-20 出版日期:2004-12-20
通讯作者: 叶淑君（1974-），女，湖北浠水人，讲师，主要从事水文地质、环境地质教学与科研工作. E-mail:E-mail：sjye@nju.edu.cn
基金资助:
国家自然科学基金项目“地面沉降条件下各向异性介质越流系统中水流模型研究”（编号：40172082）；国家自然科学基金重点项目“区域地面沉降模型研究”（编号：40335045）资助

APPLICATION OF MULTISCALE FINITE ELEMENT METHOD TO THREE DIMENSIONAL GROUNDWATER FLOW PROBLEMS IN HETEROGENEOUS POROUS MEDIA

YE Shujun, WU Jichun, XUE Yuqun

Department of Earth Sciences, Nanjing University, Nanjing 210093, China

Received:2004-04-09 Revised:2004-04-20 Online:2004-12-20 Published:2004-06-01

应用多尺度有限单元法模拟非均质多孔介质中的三维地下水流问题。与传统有限单元法相比，多尺度有限单元法的基函数具有能反映单元内参数变化的优点，所以这种方法能在大尺度上抓住解的小尺度特征获得较精确的解。在介绍多尺度有限单元法求解非均质多孔介质中三维地下水流问题的基本原理之后，对参数水平方向渐变垂直方向突变的非均质多孔介质中的三维地下水流和Borden实验场的三维地下水流分别用多尺度有限单元法和传统等参有限单元法进行了计算，结果表明在模拟高度非均质多孔介质中的三维地下水流问题时，多尺度有限单元法比传统有限单元法有效，既节省计算量又有较高的精度；在模拟非均质性弱的多孔介质中的三维地下水流问题时，多尺度有限单元法虽然也能在大尺度上获得较为精确的解，但效果不明显。

关键词: 多尺度有限单元法, 非均质, 多孔介质, 三维地下水流数值模拟

The multiscale finite element method (MsFEM) is applied to 3-D groundwater flow problems in heterogeneousporous media with different change in coefficients in the paper. The method can efficiently capture the large scale behavior of the solution without resolving all the small scale features by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator, which offers significant savings in CPU time and computer memory. The characteristic difference between MsFEM and the conventional finite element method(FEM) is attributed to base function. The base functions of MsFEM can indicate the variation of coefficients in an element, but those of the conventional FEM can't do it. The principle of the application of the multiscale finite element method to 3-D groundwater flow problems is introduced. Then two three dimensional groundwater flow problems in heterogeneous porous media are analyzed by the multiscale finite element method and the conventional finite element method, respectively. One is the 3-Dgroundwater flow problem with gradual change in coefficients in the horizontal direction and with abrupt change in the vertical direction. Another is the 3-Dgroundwater flow problem with the observation values of coefficients from the Borden test site. The solutions based on the MsFEM are much more accurate than those based on the conventional FEM with the same mesh size for the firstproblem with highly oscillatorycoefficients. The solutions based on the MsFEM are a littlemore accurate than those based on the conventional FEM with the same mesh size for the secondproblem with little change in coefficients. The applications demonstrate that the advantages of the multiscale finite element method for numerical simulation of 3-D groundwater flow in highly heterogeneous porous media, i.e. significantly reducing computational efforts, and improving the accuracy of the solutions, and that the multiscale finite element method for numerical simulation of 3-D groundwater flow in relative homogeneous porous media is effective but not obvious.

Key words: Multiscale finite element method, Heterogeneous, Porous media, Numerical simulation of 3-D groundwater flow.

中图分类号:

P641

[1]Hantush M S, Jacob C E. Nonsteady radial flow in an infinite leaky aquifer[J].EOS Transaction American Geopysical Union, 1955, 36:96-100.
[2]Hantush M S.Modification of the theory of leaky aquifers[J].Journal of Geophysical Research,1960, 65:3 713-3 726.
[3]Neuman S P, Witherspoon P A. Applicability of current theories of flow in leaky aquifers[J].Water Resources Research,1969, 5(4): 817-829.
[4]Ye Shujun(叶淑君), Dai Shuihan(戴水汉). Comparing of results of two dimensional, quasi three dimensional and three dimensional models for groundwater[J]. Hydrogeology & Engineering Geology(水文地质与工程地质)，2003，30（5）：23-27(in Chinese).
[5]Hou T Y, Wu X H. A multiscale finite element method for elliptic problems in composite materials and porous media [J].Journal of Computational Physics, 1997, 134 : 169-189. 
[6]Hou T Y, Wu X H, Cai Z. Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients[J].Mathematics of Computation,1999, 68(227): 913-943. 
[7]Cruz M E, Petera A. A parallel monte-carlo finite-element procedure for the analysis of multicomponent random media[J].International Journal for Numerical Methods in Engineering, 1995, 38:1 087-1 121.
[8]Dykaar B B, Kitanidis P K. Determination of the effective hydraulic conductivity for heterogeneous porous media using a numerical spectral approach:1. method[J].Water Resources Research, 1992, 28(4): 1 155-1 166.
[9]Durlofsky L J. Representation of grid block permeability in coarse scale models of randomly heterogeneous porous-media[J].Water Resources Research, 1992, 28: 1 791-1 800.
[10]McCarthy J F. Comparison of fast algorithms for estimating large-scale permeabilities of heterogeneous media [J].Transport in Porous Media,1995,19: 123.
[11]Babuska I, Szymczak W G. An error analysis for the finite element method applied to convection-diffusion problems [J].Computer Methods in Applied Mechanics and Engineering, 1982, 31: 19.
[12]Babuska I, Osborn E. Generalized finite element methods: Their performance and their relation to mixed methods [J].SIAM Journal on Numerical Analysis, 1983, 20: 510-536.
[13]Babuska I, Caloz G, Osborn E. Special finite element methods for a class of second order elliptic problems with rough coefficients [J]. SIAM Journal on Numerical Analysis, 1994, 31: 945-951.
[14]Ye S J, Xue Y Q, Wu J C,et al. Application of the Multiscale Finite Element Method to Ground Water Flow in Heterogeneous Porous Media [C].Proceeding of the Computational Methods in Water Resources 2004 International Conference(in press).

[1]	覃荣高, 曹广祝, 仵彦卿. 非均质含水层中渗流与溶质运移研究进展 ^*[J]. 地球科学进展, 2014, 29(1): 30-41.
[2]	王良书;刘绍文;李成;李华;徐鸣洁;钟锴;韩用兵. 岩石圈热—流变结构与大陆动力学[J]. 地球科学进展, 2004, 19(3): 382-386.

阅读次数

全文

摘要