## 2 Mathematical Model  
Traditionally, single-porosity models [7; 10], dual-porosity models [22; 23], and triple-porosity models (dual fracture models) [24; 25] are used to describe gas flow in shale reservoirs. However, single-porosity and dual-porosity models assume that the pore and fracture properties are uniform across the porous medium, while actual shale reservoirs include multiscale pores and fractures with different properties. In this case, a dual fracture model is used to capture the heterogeneity of the reservoir by dividing the porous mediums into classes and assigning each one different property [6]. As described above, the scale of the pores and fractures affect gas flow mechanisms in shale reservoirs. Fractured reservoirs have significant variation in the scale of pores and fractures. Therefore, it is reasonable to construct a dual fracture model for describing gas flow mechanisms in shale reservoirs based on the effects of multiscale pores and fractures on gas flow. The effects of multiscale pores and fractures on gas flow can be summarized as follows: (1) gas flow mechanisms for different scale pores and fractures vary with respect to the reservoir pressure; and (2) gas flow regimes at fixed scale pores and fractures varied with changes in reservoir pressure. Gas flow regimes of bulk phase gas through porous media are classified into four types based on the Knudsen number \(K_{n}\), namely, the continuum regime (\(K_{n}<0.01\)), the slip flow regime (\(0.01<K_{n}<0.1\)), the transition regime (\(1<K_{n}<10\)), and the free molecular regime (\(K_{n}>10\)). The Knudsen number is defined as follows [8; 26]:  
\[K_{n}=\frac{\lambda}{d_{e}}, \tag{1}\]  
where \(d_{e}\) is the characteristic length of the flow geometry, considered to be the effective pore diameter, and \(\lambda\) is the mean free path of the gas molecules, given by [27]:  
\[\lambda=\frac{K_{B}T}{\sqrt{2}\pi\delta^{2}p}, \tag{2}\]  
where \(T\) is the temperature, \(d_{0}\) is the diameter of a molecule of methane, \(k_{B}\) is the Boltzmann constant, and \(p\) is the reservoir pressure. Various bulk phase gas flow regimes may occur in shale reservoirs [7]. When gas is produced from a shale reservoir, the bulk phase gas in the hydraulic fractures with large scales will flow out firstly. Then the bulk phase gas in the pores and fractures in the matrix with small scales will transfer into hydraulic fractures. The adsorption/desorption process of adsorbed phase gas occurs with changes in gradient concentration along the pore walls with nanoscales, resulting in surface diffusion. Bulk phase gas and adsorbed gas in the pores and fractures of matrix supply the shale gas production. When the gas is produced from the reservoir, the gas pressure will decrease to a lower magnitude, and effective stress will be changed. This variation of effective stress will affect the average pores radius of the matrix, which means that the intrinsic permeability is a variable parameter [6]. When reservoir pressure decreases, the gas flow regimes of bulk phase gas in different scale pores and fractures are transformed from the continuum regime to the slip flow regime, and finally to the transition regime. The free molecular regime only occurs in nanopores at low pressures (Figure 1). Under geological conditions, the transition and free molecular regimes are mainly observed in nanoscale pores, the slip flow regime is mainly observed in microscale pores, and the continuum regime is mainly observed in microscale pores/fractures and macroscale fractures. Based on the dual fracture model [25], a shale reservoir was divided into two sections, namely, hydraulic fractures and matrices (between hydraulic fractures) with well-developed nano-micro-scale pores and micro-macroscale natural fractures. Both bulk phase gas flow and adsorbed gas flow occur within the shale matrix. However, only bulk phase gas flow occurswithin hydraulic fractures. Fracture properties play an important role in reservoir performance in the early life of a shale gas well.