In this paper, we propose a multiscale method for solving the Darcy flow of a single-phase fluid in two-dimensional fractured porous media. We consider a discrete fracture-matrix (DFM) model that treats fractures as one-dimensional objects, and flows in both the matrix and fractures respect the Darcy’s law. A multipoint flux mixed finite element (MFMFE) method with the broken Raviart–Thomas (${\text{RT}}^{\frac{1}{2}}$) element and the trapezoidal quadrature rule is employed to approximate the matrix velocity and pressure, which results in a block diagonal, symmetric and positive definite mass matrix for the matrix velocity on general quadrilateral grids; the one-dimensional ${\text{RT}}_0$ mixed finite element method with the one-dimensional trapezoidal quadrature rule is exploited to approximate the fracture velocity and pressure, which leads to a diagonal and positive definite mass matrix for the fracture velocity in each single fracture. All these features of the obtained mass matrices allow for velocity elimination. Multiscale basis functions are constructed for the two-dimensional matrix pressure following the generalized multiscale finite element method (GMsFEM) framework to capture the fine-scale information of heterogeneous fractured porous media and effectively reduce the degrees of freedom for the matrix pressure, while fine-grid basis functions are utilized for the one-dimensional fracture pressure in fractures. Various numerical tests with the oversampling technique for different fracture distributions are performed to show that the proposed multiscale method is effective and able to provide good approximations for the fine-grid solution.

在本文中，我们提出了一种多尺度方法来求解二维裂缝性多孔介质中单相流体的达西流动。我们考虑离散裂缝矩阵（DFM）模型，该模型将裂缝视为一维对象，并且在矩阵和裂缝中的流动都遵循达西定律。断裂Raviart–Thomas的多点通量混合有限元（MFMFE）方法（${\text{逆时针}}^{\frac{\mathrm{1个}}{\mathrm{2个}}}$）元素，并使用梯形正交规则来近似矩阵速度和压力，这将为一般四边形网格上的矩阵速度生成块对角线，对称和正定质量矩阵；一维的${\text{逆时针}}_0$利用一维梯形正交规则的混合有限元方法来逼近断裂速度和压力，从而得出每个单条断裂速度的对角正定质量矩阵。所获得的质量矩阵的所有这些特征都可以消除速度。遵循广义多尺度有限元方法（GMsFEM）框架，为二维基质压力构造多尺度基函数，以捕获非均质裂隙多孔介质的精细尺度信息，并有效降低基质压力的自由度，而网格基函数用于一维骨折中的骨折压力。使用过采样技术对不同的裂缝分布进行了各种数值测试，结果表明所提出的多尺度方法是有效的，并且能够为精细网格解决方案提供良好的近似值。