The level of detail on modern geological models requires higher resolution grids that may render the simulation of multiphase flow in porous media intractable. Moreover, these models may comprise highly heterogeneous media with phenomena taking place in different scales. Scale transferring techniques allow for the solution of such problems in a lower resolution scale at reduced computational cost. Among these techniques, the Multiscale Finite Volume (MsFV) method constructs a set of numerical operators in order to map quantities from the fine-scale mesh to a coarser one and vice-versa while maintaining flux conservation on both scales. However, the MsFV formulation, as originally stated, is only consistent on k-orthogonal grids since it uses a linear Two-point Flux Approximation (TPFA) method and may struggle to generate consistent primal-dual coarse grids pairs on unstructured grids. The Multiscale Restriction Smoothed-Basis method (MsRSB) improves on the MsFV by introducing a new iterative procedure to find the multiscale operators and the concept of support regions which reduces the method's complexity when applied to unstructured fine and coarse grids. The original version was only consistent on k-orthogonal fine grids due to the TPFA discretization, but filtering methods have been developed to also enable consistent multipoint schemes on the fine scale. Meanwhile, the Multiscale Control Volume method (MsCV) replaces the TPFA by the Multipoint Flux Approximation with a Diamond stencil (MPFA-D) scheme on the fine-scale while further enhancing the generation of the geometric entities to allow unstructured grids on the fine and coarse scales for two-dimensional simulations. In this work we propose an extension to three-dimensional geometries of both the MsCV and the algorithm to obtain the multiscale geometric entities based on the concept of a background grid, a coarser grid used as a proxy for the primal coarse grid. We modify the original MPFA-D method to use the very robust Global Least Squares (GLS) interpolation technique to obtain the required auxiliary nodal unknowns. We also introduce an enhanced version of the 3-D MsCV with the incorporation of the enhanced MsRSB (E-MsRSB) to enforce M-matrix properties and improve convergence. Finally, we employ the MsCV operators in a two-stage smoother to show how it can be used as a good iterative procedure to recover the fine-scale solution. We show that the 3-D MsCV method produces good results employing unstructured grids on both scales to handle the simulation of the single-phase flow in anisotropic, and heterogeneous porous media and that the smoothing procedure is able to accurately retrieve the fine-scale solution within a few iterations.

中文翻译：

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用于模拟各向异性和非均质多孔介质中单相流的多尺度控制体积方法的 3-D 扩展


现代地质模型的详细程度需要更高分辨率的网格，这可能会使多孔介质中多相流的模拟变得棘手。此外，这些模型可能包含高度异构的介质，其中现象发生在不同的尺度上。尺度转移技术允许以较低的分辨率尺度以降低的计算成本解决此类问题。在这些技术中，多尺度有限体积 (MsFV) 方法构造了一组数值算子，以便将精细尺度网格中的量映射到较粗尺度网格，反之亦然，同时在两个尺度上保持通量守恒。然而，正如最初所述，MsFV 公式仅在 k 正交网格上一致，因为它使用线性两点通量近似 (TPFA) 方法，并且可能难以在非结构化网格上生成一致的原始-对偶粗网格对。多尺度限制平滑基方法 (MsRSB) 通过引入新的迭代过程来查找多尺度算子和支持区域的概念，从而在 MsFV 的基础上进行了改进，从而降低了该方法在应用于非结构化精细和粗糙网格时的复杂性。由于 TPFA 离散化，原始版本仅在 k 正交精细网格上保持一致，但已开发出滤波方法以在精细尺度上实现一致的多点方案。同时，多尺度控制体积方法（MsCV）在精细尺度上用钻石模板多点通量近似（MPFA-D）方案取代了TPFA，同时进一步增强了几何实体的生成，以允许精细尺度上的非结构化网格二维模拟的粗尺度。 在这项工作中，我们提出了对 MsCV 和算法的三维几何的扩展，以基于背景网格的概念获得多尺度几何实体，背景网格是用作原始粗网格的代理的较粗网格。我们修改原始的 MPFA-D 方法，使用非常鲁棒的全局最小二乘 (GLS) 插值技术来获得所需的辅助节点未知数。我们还引入了 3-D MsCV 的增强版本，其中结合了增强型 MsRSB (E-MsRSB)，以增强 M 矩阵属性并提高收敛性。最后，我们在两级平滑器中使用 MsCV 算子，以展示如何将其用作良好的迭代过程来恢复精细尺度解。我们表明，3-D MsCV 方法在两个尺度上采用非结构化网格来处理各向异性和非均质多孔介质中的单相流模拟，产生了良好的结果，并且平滑过程能够准确地检索精细尺度解在几次迭代内。