Modified Multiscale Finite Element Method For 2-D Groundwater Flow Problems

Most groundwater aquifer systems are heterogeneous, and the heterogeneity of the media often spans over many scales. When solving groundwater problems in het-erogeneous media, the linear finite element method (LFEM) and the finite difference method (FDM) need a very fine grid to ensure the hydraulic conductivity can be regard as constant in each element. If the study region is very large, the fine grid of the LFEM and FDM leads to a great amount of computational cost. Therefore, water researchers devote themselves to find methods which can not only decrease the number of element but also achieve accurate results. The multiscale finite element method (MSFEM) is one of such methods.In 1997, Hou and Wu introduce the multiscale finite element method (MSFEM) for solving groundwater problems in heterogeneous media. The core of the MSFEM is the construction of base functions by solving the reduced locally problem so as to capture the fine-scale information. Using these base functions, the MSFEM can solve heads directly in the coarse scale grid, without solving the small scale feathers. There-fore, the MSFEM is more efficient than LFEM and FDM in solving heterogeneous groundwater problems. However, for the large-scale, long-term or complex condition groundwater flow problems, the MSFEM needs much computational cost to construct base functions, which is not efficiency. On the other hand, the MSFEM can’t obtain continuous heads derivatives, so that the accuracy of MSFEM for solving velocity is low. In order to improve the the efficiency and accuracy of MSFEM, the topic of MS-FEM modification was focused on.This dissertation contains two parts. In the first part, the dissertation intends to find a way to improve the efficiency of MSFEM base function construction. Based on the principle of base functions of capturing fine scale information, the dissertation introduces a modified coarse element subdivision for MSFEM. Compare to the orig-inal coarse element subdivision, the modified one needs much less interior nodes to divide a coarse element into the same number fine elements, which ensures the amount of fine scale information. Due to the unknowns of base functions is determined by the interior nodes number of coarse element, the modified subdivision can save much computational costs.Based on the modified subdivision, the dissertation introduces a modified mu-tilscale finite element method (MMSFEM). The dissertation uses 7 different ground-water problems to exam the accuracy and efficiency of the MMSFEM. The results shows that, compare to the MSFEM, the MMSFEM can save more than 90% compu-tational cost and achieves nearly the same accuracy. Meanwhile, the dissertation finds out that the MMSFEM is also sensitive to the boundary condition of base functions, oscillatory boundary condition can improve the accuracy of MMSFEM. On the other hand, the modified coarse element subdivision is applied to the upscaling method so as to improve the efficiency of solving upscaling coefficient.Continuous nodal Darcian velocity is important for the water resources assess-ment and advection-dispersion equation. However, same as the LFEM, the MSFEM can not obtain accurate and continuous nodal Darcian velocity. Therefore, the second part of this dissertation focus on the ability of solving continuous Darcian velocity of MSFEM. Therefore, the dissertation introduces a cubic-spline mutiscale finite element method(CMSFEM). The heads computation process of the CMSFEM is same as the MSFEM, which is more efficiency than LFEM and FDM. Then the CMSFEM employs the cubic-spline technique to ensure the continuity of head derivatives, so the Darcian velocity can be continuous. The CMSFEM can solve the Darcian velocities not only at coarse scale grid nodes but also at the nodes in the fine scale grid. The computa-tion of coarse-scale velocities is similar to that of Zhang’s cubic-spline method. The CMSFEM breaks the full study region problem down to local coarse elements so that the process of solving the fine-scale velocities is decoupled from element to element, which saves much computational costs. In the numerical simulations, the CMSFEM is compared with Yeh’s Galerkin model and Zhang’s cubic-spline method. The results shows that the CMSFEM needs much less CPU time than Yeh and Zhang’s method, while ensuring computational accuracy.At last, some analytic results of the dissertation concluded, the main conclusion of MMSFEM and CMSFEM are introduced. In addition, the dissertation gives some suggestions on the future studies of MSFEM.