A Stokes-Darcy-Darcy model and its discontinuous Galerkin method on polytopic grids
A Stokes-Darcy-Darcy model and its discontinuous Galerkin method on polytopic grids
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DOI：
10.1016/j.jcp.2024.112780
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发表时间：
2024-03
期刊：
影响因子：
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通讯作者：
Rui Li;Yali Gao;Chen-Song Zhang;Zhangxin Chen
中科院分区：
文献类型：
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作者：
Rui Li;Yali Gao;Chen-Song Zhang;Zhangxin Chen
在本文中，我们提出并数值求解了一个模型，该模型考虑了裂缝多孔介质中的受限流动以及管道中的自由流动。例如，当流体在裂缝性孔洞碳酸盐岩储层和洞穴/管道中流动时，就会出现这种情况，其中基质和裂缝是流体储存和流动的主要空间。洞穴和管道中的流动受斯托克斯方程控制；由基质和裂缝组成的裂缝性多孔介质中的流动由耦合达西-达西（块体-裂缝）模型描述。然后，这两个模型通过裂缝多孔介质和洞穴/管道之间的界面上合适的（物理上一致的）界面条件进行耦合，在实现高精度的物理忠实模拟中发挥着关键作用。耦合模型的所有界面条件的构造都是基于传统体断裂模型和著名的斯托克斯-达西模型的基本性质。推导了该模型的弱公式，并对模型的适定性进行了理论分析。我们还针对管道中的斯托克斯流和多孔基体中的达西流提出了多面体不连续伽辽金有限元近似，该近似与裂缝中流动的一致有限元方案相结合。获得合适能量范数下的最佳误差估计。进行二维数值实验来验证所提出的模型，并展示模型和数值方法的特征，例如数值解的最佳收敛速度；裂缝、基质和管道周围的详细流动特性；以及对现实问题的适用性。
In this paper, we propose and numerically solve a model considering confined flow in fractured porous media coupled with free flow in conduits. Such a situation arises, for example, for fluid flows in fractured vuggy carbonate reservoirs and caves/conduits, where a matrix and fractures are the main spaces for fluid storage and flow. The flow in caves and conduits is governed by the Stokes equation; the flow in fractured porous media, which consists of both a matrix and fractures, is described by a coupled Darcy–Darcy (bulk–fracture) model. Then, the two models are coupled through suitable (physically consistent) interface conditions on the interface between fractured porous media and caves/conduits, playing a key role in a physically faithful simulation that achieves high accuracy. The construction of all the interface conditions of the coupled model is based on the fundamental properties of the traditional bulk–fracture model and the well-known Stokes–Darcy model. The weak formulation is derived for the proposed model, and the well-posedness of the model is theoretically analyzed. We also propose a polyhedral discontinuous Galerkin finite element approximation for the Stokes flow in conduits and Darcy flow in the porous matrix, which is coupled with a conforming finite element scheme for the flow in the fracture. Optimal error estimates in a suitable energy norm are obtained. Two-dimensional numerical experiments are conducted to validate the proposed model and to demonstrate the features of both the model and the numerical method, such as the optimal convergence rate of the numerical solution; the detail flow characteristics around fractures, the matrix, and conduits; and the applicability to real-world problems.