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Modelling of highly-heterogeneous media using a flux-vector-based Green element method

机译：使用基于通量向量的Green元素方法对非均质介质进行建模

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摘要

One of the modern techniques for solving nonlinear problems encountered in flow in porous media is the Green element method (GEM). It combines the high accuracy of the boundary element method with the efficiency and versatility of the finite element method. The high accuracy of the GEM comes from the direct representation of the normal fluxes as unknowns. However, in the classical GEM procedure the difficulties imposed by a large number of normal fluxes at each internal node are typically overcome by approximating them in terms of the primary variable, and this can lead to a diminution of the overall accuracy, particularly in applications to heterogeneous media. To maintain the high accuracy, another approach was proposed, namely the 'flux-vector-based GEM', or 'q-based' GEM. According to this approach, only two and three unknown flux components are required for a node in two- and three-dimensional domains, respectively. An important advantage of this approach is that the flux vector at each node is determined directly. We present a comparison between the results obtained using the classical GEM and those obtained using the 'q-based' GEM for problems in heterogeneous media with permeability changes of several orders of magnitude. In some simple examples of such situations, the classical GEM fails to produce physically sensible results, whilst our novel development of the 'q-based' GEM is in general agreement with both the boundary element method and the control volume method. However, in order to acquire results to the same degree of accuracy, these latter two approaches are less efficient than the 'q-based' GEM. This new approach is thus suitable for overcoming the difficulties present when modelling flow in heterogeneous media with rapid and high-order changes in material parameters. Within geological problems, we can therefore apply it to flow through layered sequences, across partially-sealing faults and around wells. The 'q-based' GEM approach presented here is developed and applied for rectangular grids, and we provide details of the extension to triangular finite-element-type meshes in two dimensions.
机译：解决多孔介质中流动中遇到的非线性问题的现代技术之一是格林元素方法（GEM）。它结合了边界元方法的高精度与有限元方法的效率和多功能性。 GEM的高精度来自将正常磁通量直接表示为未知量。但是，在经典GEM程序中，通常通过按一次变量对它们进行近似来克服由每个内部节点处的大量法向通量所带来的困难，这可能导致总体精度降低，尤其是在应用到异构媒体。为了保持高精度，提出了另一种方法，即“基于磁通矢量的GEM”或“基于q的” GEM。根据这种方法，在二维和三维域中的节点分别仅需要两个和三个未知通量分量。这种方法的一个重要优点是，可以直接确定每个节点处的磁通矢量。我们对使用经典GEM获得的结果与使用“基于q的” GEM获得的结果进行比较，以解决渗透率变化几个数量级的非均质介质中的问题。在这种情况的一些简单示例中，经典GEM无法产生物理上有意义的结果，而我们基于q的GEM的新颖开发与边界元法和控制量法大体一致。但是，为了获得相同精度的结果，后两种方法的效率不如基于“ q”的GEM。因此，这种新方法适用于克服在材料参数快速高阶变化下对异质介质中的流体进行建模时遇到的困难。因此，在地质问题中，我们可以将其应用于层状序列，部分封闭断层和井周围的流动。本文介绍的“基于q”的GEM方法已开发并应用于矩形网格，并且我们提供了二维二维三角形有限元类型网格扩展的详细信息。

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