## 2 Mathematical Model
However, it is the properties of the matrices that control the gas flow behavior over longer periods of time [17]. This study takes the Marcellus Shale as an example. The study area (Marcellus) is located in the eastern United Three major orogenic events affected the sedimentation and structure in the basin during the Paleozoic Era. From oldest to youngest these are the Taconic orogen (Ordovician), the Acadian orogen (Devonian to Early Mississippi), and the Alleghanian orogen (Mississippian to Permian) [28]. The Marcellus Shale, which is Devonian age (416-359.2 Ma), was deposited during a period of rapid transgression in anoxic to euxinic waters within a restricted foreland basin. The Marcellus Shale is characterized by its organic-rich nature and black color, and is integrated with the lower Onondaga Limestone and the upper Tully Limestone [29]. The vitrinite reflectance (Ro) values between 2% and 2.2%, indicating that the maturity of shale is mainly in over-mature stages [30], and are dominantly in the thermogenic generation stage. The calculation procedures include five stages, which are shown in Figure 2. First, a suitable mathematical model will be applied for describing gas flow in multiscale shale reservoirs. Second, flow boundaries (pore scales) of various flow regimes will be obtained for a given condition with considering coupled effects of multiscale pores/fractures, adsorption/desorption, and stress-sensitivity. Third, individual flow models for various flow regimes will be proposed. Fourth, those individual flow models will be verified by comparing them with traditional models and matching the production data. Further, the coupled model obtained by combining those individual flow models would be applied for modeling gas flow in the research area. The reservoir characteristic parameters and the field production data of a shale gas well, which was located between the lower section and upper section of the Marcellus formation, were chosen to verify the proposed models and conduct the further analysis. The initial reservoir conditions of the Marcellus Shale well were characterized by a deeper vertical depth of 2619.1 m, a high reservoir temperature of 352.55 K, an overpressure condition with a high initial reservoir pressure of 3.26 \(\times\) 10\({}^{7}\) P, and a high gas saturation of 0.70, but low initial matrix permeability (0.0006 mD) and porosity (6.5%) [28]. In addition, the two main adsorption parameters (Langmuir pressure and Langmuir volume) of Marcellus Shale were an average of 2.5 m\({}^{3}\)/t and 3.0 \(\times\) 10\({}^{7}\) Pa, which illustrates that the Marcellus Shale had good adsorption capability [6]. Figure 1: Mathematical model for describing gas flow in multiscale shale reservoirs. In order to successfully achieve the aims of this study, the reservoir parameters above were basic parameters for the simulation of gas flow in multiscale shale reservoirs. In addition, other parameters such as gas viscosity, fracture permeability, surface diffusion coefficient, drainage area, and horizontal well length need to be obtained. Fortunately, a lot of research has been conducted on gas flow in the Marcellus Shale, and appropriate reservoir parameters have been given by previous studies [6; 28; 31]. The initial reservoir pressure of shale gas development sections is dominated by <60 MPa. The pore scale boundaries of various flow regimes vary with the change in reservoir pressure, and are more sensitive response to the low reservoir pressure (<5 MPa) (Figure 1). Considering the initial geological conditions and the response of gas flow boundary to reservoir pressure, the initial reservoir pressure is set to 60 MPa, and the reservoir pressure is discretized as follows: the simulated pressure steps of reservoirs pressure between 5 and 60 MPa, and <5 MPa, are 5 MPa and 1 MPa, respectively. Then the boundaries of various flow regimes for a certain reservoir pressure would be obtained.