A finite difference approximation of reduced coupled model for slightly compressible Forchheimer fractures in Karst aquifer system

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.