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

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

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。

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.