Two‐phase flows of oil and water in naturally fractured reservoirs can be described by a system of nonlinear partial differential equations that comprises of an elliptic pressure equation and hyperbolic saturation equation coupled through the total velocity field. Modeling this problem is a great challenge, due to the complexity of the depositional environments, including inclined layers and fractures with different sizes and shapes, and random spatial distribution. In this work, to solve the pressure equation, we adopted a cell‐centered finite‐volume method with a multipoint flux approximation that uses the “diamond stencil” (MPFA‐D) coupled with a hybrid‐grid method (HyG) to deal with the fractures. The classical first‐order upwind method was used to solve the saturation equation, in its explicit and implicit versions. The MPFA‐D is a very robust and flexible formulation that is capable of handling highly heterogeneous and anisotropic domains using general polygonal meshes. In the strategy developed in this work, the mesh that discretize the domain must fit the spatial position of the fractures, so that they are associated to the control surfaces—as (n − 1)D cells—therefore, the calculation of the fluxes in these control surfaces is dependent on the pressures on fractures and on the adjacent volumes. In HyG, the fractures are expanded to nD in the computational domain. The proposed formulation presented quite remarkable results when compared with similar formulations using classical full pressure support and triangle pressure support methods, or even the with MPFA‐D itself when the fractures are treated as nD geometric entities.

中文翻译：

---

金刚石网格有限体积方案的多点通量逼近，用混合网格方法二维模拟自然裂缝性储层中的流体流动

天然裂隙油藏中油和水的两相流可以通过非线性偏微分方程组来描述，该系统由通过总速度场耦合的椭圆压力方程和双曲线饱和方程组成。由于沉积环境的复杂性（包括具有不同大小和形状的倾斜层和裂缝以及随机的空间分布），对此问题进行建模是一个巨大的挑战。在这项工作中，为解决压力方程，我们采用了以单元为中心的有限体积方法，该方法具有多点通量近似值，该方法使用“菱形模板”（MPFA-D）和混合网格方法（HyG）来处理骨折。使用经典的一阶迎风方法来求解饱和方程式的显式和隐式形式。MPFA-D是一种非常强大且灵活的公式，它能够使用常规的多边形网格处理高度异构和各向异性的区域。在这项工作制定的策略中，离散化区域的网格必须适合裂缝的空间位置，以便将它们与控制面相关联，如（n − 1）D个单元-因此，这些控制面中的通量的计算取决于裂缝和相邻体积上的压力。在HyG中，裂缝在计算域中扩展为n D。与使用经典全压力支撑和三角压力支撑方法的相似公式相比，或与将裂缝视为n D几何实体的MPFA-D本身相比，拟议的公式显示出相当出色的结果。