On continuous and discontinuous approaches for modeling groundwater flow in heterogeneous media using the Numerical Manifold Method: Model development and comparison
On continuous and discontinuous approaches for modeling groundwater flow in heterogeneous media using the Numerical Manifold Method: Model development and comparison
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DOI：
10.1016/j.advwatres.2015.03.004
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发表时间：
2015-06
影响因子：
4.7
通讯作者：
M. Hu;Yuan Wang;J. Rutqvist
中科院分区：
文献类型：
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作者：
M. Hu;Yuan Wang;J. Rutqvist
在非均质地质介质中模拟地下水流动的一个主要挑战是模拟任意定向或相交的边界和内部材料界面。数值流形法（NMM）由于其处理边界的能力、构造物理覆盖函数（连续或有梯度跳跃）的灵活性、与固定数学网格（覆盖）的网格划分效率、提高近似精度的便利性以及通过单纯形积分实现的积分精度，最近成为这种建模的一种有前途的方法。在本文中，我们报告了两种新的边界约束方法的发展和比较，即具有跳跃函数的连续方法和具有拉格朗日乘子的不连续方法。在不连续拉格朗日乘子法中，将材料界面视为不连续点，将数学覆盖划分为不同的物理覆盖。我们定义并推导了拉格朗日乘子的严格形式来连接划分的物理覆盖，从而满足了折射定律的连续性要求。在连续跳跃函数法（JFM）中，将材料界面视为包含在物理覆盖物中的内部界面。为了满足折射定律，我们简单地定义了跳跃项来表示横过界面的头部梯度的不连续。然后从全局自由度、多材料界面处理、小面积处理、运动界面处理、与力学分析耦合的可行性以及对其他数值方法的适用性等方面对两种方法进行了理论比较。将新导出的边界约束方法编码到地下水流量分析的NMM模型中，并在不同的模拟实例上进行了精度和效率的测试。我们首先测试了LMM的Dirichlet边界，然后测试了LMM和JFM的理想异质模型，并将数值结果与解析解进行了比较。然后，我们在一个非均匀模型中测试了这两种方法，并比较了水头和比流量的结果。考虑到边界约束的高精度、处理任意方向或复杂相交边界的能力以及使用固定数学网格的效率，这两种方法都适用于材料边界的建模。
One major challenge in modeling groundwater flow within heterogeneous geological media is that of modeling arbitrarily oriented or intersected boundaries and inner material interfaces. The Numerical Manifold Method (NMM) has recently emerged as a promising method for such modeling, in its ability to handle boundaries, its flexibility in constructing physical cover functions (continuous or with gradient jump), its meshing efficiency with a fixed mathematical mesh (covers), its convenience for enhancing approximation precision, and its integration precision, achieved by simplex integration. In this paper, we report on developing and comparing two new approaches for boundary constraints using the NMM, namely a continuous approach with jump functions and a discontinuous approach with Lagrange multipliers. In the discontinuous Lagrange multiplier method (LMM), the material interfaces are regarded as discontinuities which divide mathematical covers into different physical covers. We define and derive stringent forms of Lagrange multipliers to link the divided physical covers, thus satisfying the continuity requirement of the refraction law. In the continuous Jump Function Method (JFM), the material interfaces are regarded as inner interfaces contained within physical covers. We briefly define jump terms to represent the discontinuity of the head gradient across an interface to satisfy the refraction law. We then make a theoretical comparison between the two approaches in terms of global degrees of freedom, treatment of multiple material interfaces, treatment of small area, treatment of moving interfaces, the feasibility of coupling with mechanical analysis and applicability to other numerical methods. The newly derived boundary-constraint approaches are coded into a NMM model for groundwater flow analysis, and tested for precision and efficiency on different simulation examples. We first test the LMM for a Dirichlet boundary and then test both LMM and JFM for an idealized heterogeneous model, comparing the numerical results with analytical solutions. Then we test both approaches for a heterogeneous model and compare the results of hydraulic head and specific discharge. We show that both approaches are suitable for modeling material boundaries, considering high accuracy for the boundary constraints, the capability to deal with arbitrarily oriented or complexly intersected boundaries, and their efficiency using a fixed mathematical mesh.