A finite difference method is proposed for solving the compressible reduced coupled model, in which the flow is governed by Forchheimer’s law in the fracture and Darcy’s law in the surrounding porous media. By using the averaging technique, the fracture is reduced to a lower dimensional interface and a more complicated transmission condition is derived on the fracture-interface. Different degrees of freedom are located on both sides of fracture-interface in order to capture the jump of velocity and pressure. Second-order error estimates in discrete norms are derived on nonuniform staggered grids for both pressure and velocity. The proposed scheme can also be extended to nonmatching spatial and temporal grids without loss of accuracy. Numerical experiments are performed to demonstrate the efficiency and accuracy of the numerical method. It is shown that the parameter ξ has little influence on the fluid flow, and the permeability tensor of fracture has a significant impact on the flow rate in both the surrounding porous and fracture-interface.

提出了一种有限差分方法来求解可压缩的简化耦合模型，该模型中的流动由裂缝中的福希海默定律和周围多孔介质中的达西定律控制。通过使用平均技术，可以将裂缝减小到较低尺寸的界面，并在裂缝界面上导出更复杂的传输条件。不同的自由度位于裂缝界面的两侧，以便捕获速度和压力的跃变。离散范数中的二阶误差估计是在压力和速度的非均匀交错网格上得出的。所提出的方案还可以扩展到不匹配的空间和时间网格，而不会损失准确性。进行数值实验以证明数值方法的效率和准确性。ξ对流体流动的影响很小，裂缝的渗透率张量对周围的孔隙和裂缝界面的流速都有很大的影响。