# A Coupled Darcy-Forchheimer Flow Model in Fractured Porous Media
## 1 Introduction  
Seepage in the rock mass is one of the cores of rock mechanics, involving tunneling, mining, petroleum, hydropower, geothermal exploitation, and other projects [1, 2, 3]. On the one hand, the high permeability of rock fracture leads to a higher fluid flow rate in the fracture compared with the flow in the rock matrix [4, 5]. On the other hand, rock permeability is low, and a fluid will pass slowly through the matrix in a rock mass. A complex coupling process for fluid flow exists in fracture and matrix. Therefore, it is still challenging to establish an effective seepage model for rock mass. Since the 1970s, various seepage models of rock mass have been proposed based on the understanding of fracture characteristics and distribution [6, 7, 8]. The continuum media model is applicable to model fluid flow in a rock mass with a large number of small and regular distribution fractures [9, 10]. Based on the homogeneity theory, seepage coefficient tensor theory, generalized Darcy's law, and the fluid continuity equation, the partial differential equations of flow and pressure are derived. The partialdifferential equations are solved by the initial conditions and the boundary, and the velocity and pressure fields of the whole flow field are obtained. However, when there are masses of faults or fractures in the study area, the rationality of using the equivalent continuum model remains to be discussed. The discrete fracture network model only considers the seepage of macroscopic fractures in the rock mass, which are usually generated randomly according to their occurrence, size, and hydrological parameters [11]. The governing equation is established based on the flow balance principle at the intersection of each fracture, and then the matrix equation of pressure and flow is obtained by combining the flow equation of a single fracture. Finally, the velocity field and pressure field are obtained by solving the equations. The discrete fracture network model is suitable for large-scale fractures with sparse distribution in the rock mass. However, it does not consider the flow condition of the rock block, and the error may be larger in some high-porosity rock masses. The dual medium model divides fractured rock mass into a porous system and a fractured system [12; 13]. It is assumed that water flows on the porous system and the porous system is used to store fluid. The two flow systems exchange on the fracture wall, which satisfies the conservation law of flow. The generalized model holds that the interconnected fracture is the channel of fluid flow, while the pore and the isolated fracture are the stores of fluid. The flow exchange between the two on the interconnected fracture wall satisfies the law of flow conservation. Therefore, the dual medium model can describe the preferential flow phenomenon. At the same time, the exchange effect of water flow is taken into account, which makes the simulation have better accuracy. In the early days, dual media models assumed that fluid flow in fractured or porous media would conform to Darcy's law. Darcy's Law is consistent with a single-phase incompressible flow in porous media at low flow rates, such as water-bearing reservoirs in an underground rock mass with low permeability. However, Darcy's Law does not apply to wells or high permeability fracture areas where flow rates are higher. To describe the seepage law at high flow rates, one of the simplest ways is to add a second amendment term in Darcy's Law, which has been widely adopted as Forchheimer's Law. The law combines the viscous effect and the inertia effect: At low flow, the viscous effect dominates, and the model is simplified to [PERSON]'s law. When the flow rate is increased, the inertia effect is dominant, which makes the pressure increase nonlinearly with the flow rate. [PERSON] and [PERSON] (2008) [14] and [PERSON] (2004) [15] proved that [PERSON]'s law for laminar flow is effective in a certain range. [PERSON] et al. (2001) [16] and [PERSON] et al.