## 3 Mathematical Model
### Flow Model for Stress-Sensitive Gas Reservoirs  
The reservoir permeability changes with reservoir pressure and can be described as follows [37]:  
\[k=k_{\mathrm{i}}e^{-\gamma(\psi_{\mathrm{i}}-\psi)}, \tag{1}\]  
where \(\psi\) is the pseudo-pressure, which is defined as  
\[\psi=\int_{0}^{p}\frac{2p}{\mu Z}\mathrm{d}p. \tag{2}\]  
The governing equation of gas flow in stress-sensitive porous media is given as follows [37]:  
\[\frac{\partial^{2}\psi}{\partial r^{2}}+\frac{1}{r}\frac{\partial\psi}{ \partial r}+\gamma\left(\frac{\partial\psi}{\partial r}\right)^{2}=e^{\gamma( \psi_{\mathrm{i}}-\psi)}\frac{\phi\mu c_{\mathrm{t}}}{k_{\mathrm{i}}}\frac{ \partial\psi}{\partial t}. \tag{3}\]  
Both gas viscosity \(\mu\) and total compressibility \(c_{\mathrm{t}}\) in Equation (3) are functions of pressure. In order to simplify the proposed model and obtain an analytical solution, many research papers [26; 37; 38] treat the values of \(\mu\) and \(c_{\mathrm{t}}\) in Equation (3) as constants, which could make the simulation results be accurate enough for engineering requirements. Following the suggestion by [PERSON] [37], the values of \(\mu\) and \(c_{\mathrm{t}}\) are evaluated at the initial condition, i.e., \(\mu=\mu(p_{\mathrm{i}})\) and \(c_{\mathrm{t}}=c_{\mathrm{t}}(p_{\mathrm{i}})\) in this paper. Figure 2: Flow chart of the solution methodology. With the assumption of the uniform initial pressure, the initial condition is expressed as  
\[\psi(r,t=0)=\psi_{\rm i}. \tag{4}\]  
The gas reservoir is assumed to be laterally infinite, and thus, the outer boundary condition is  
\[\psi(r\rightarrow\infty,t)=\psi_{\rm i}. \tag{5}\]  
The inner boundary condition for a point source can be written as  
\[\lim_{r\to 0}e^{-\gamma(\psi_{\rm i}-\psi)}r\frac{\partial\psi}{\partial r}= \frac{p_{\rm sc}q(r=0,t)T}{\pi k_{\rm i}hT_{\rm sc}}. \tag{6}\]  
Introducing the dimensionless variables listed in Table 1, Equations (3)-(6) are rewritten as  
\[\frac{\partial^{2}\psi_{\rm D}}{\partial r_{\rm D}^{2}}+\frac{1}{r_{\rm D}} \frac{\partial\psi_{\rm D}}{\partial r_{\rm D}}-\gamma_{\rm D}\Big{(}\frac{ \partial\psi_{\rm D}}{\partial r_{\rm D}}\Big{)}^{2}=e^{\gamma_{\rm D}\psi_{ \rm D}}\frac{\partial\psi_{\rm D}}{\partial t_{\rm D}}, \tag{7}\]  
\[\psi_{\rm D}(r_{\rm D},t_{\rm D}=0)=0, \tag{8}\]  
\[\psi_{\rm D}(r_{\rm D},t_{\rm D}=0)=0, \tag{9}\]  
\[\psi=\int_{0}^{p}\frac{2p}{\mu Z}\mathrm{d}p. \tag{2}\]0  
The point source model including Equations (7)-(10) shows strong non-linearity which can be alleviated by introducing the following expression [16]:  
\[\psi=\int_{0}^{p}\frac{2p}{\mu Z}\mathrm{d}p. \tag{2}\]1  
The perturbation technique, Laplace transform, and superposition principle [39] were employed to obtain the pressure distribution in the gas reservoir with a vertical fractured well with multiple hydraulic fractures producing at a constant rate (see Appendix B):  
\[\psi=\int_{0}^{p}\frac{2p}{\mu Z}\mathrm{d}p. \tag{2}\]2 \begin{table}
\begin{tabular}{c c} \hline
**Nomenclature** & **Definition** \\ \hline Dimensionless pseudo – pressure & \(\psi_{\rm D}=\frac{\pi k_{\rm i}hT_{\rm sc}}{Q_{\rm sc}p_{\rm sc}T}(\psi_{\rm i }-\psi)\) \\ Dimensionless transformed pseudo – pressure & \(\xi_{\rm D}=\frac{\pi k_{\rm i}hT_{\rm sc}}{Q_{\rm sc}p_{\rm sc}T}\pi\) \\ Dimensionless time & \(t_{\rm D}=\frac{\pi}{\phi_{\rm c}\mu_{\rm w}^{2}}\) \\ Dimensionless distance & \(r_{\rm D}=\frac{t}{\pi}\) \\ Dimensionless wellbore storage coefficient & \(C_{\rm D}=\frac{1}{2\pi\phi_{\rm c}\mu_{\rm w}^{2}}\) \\ Dimensionless flow rate & \(q_{\rm D}=\frac{q}{\phi_{\rm w}}\) \\ Dimensionless flow – rate density & \(q_{\rm D}=\frac{\partial}{\partial\xi_{\rm c}}\) \\ Dimensionless permeability modulus & \(\gamma_{\rm D}=\gamma\frac{Q_{\rm sc}p_{\rm sc}T}{\pi k_{\rm i}hT_{\rm sc}}\) \\ Fracture conductivity coefficient & \(C_{\rm D}=\frac{\pi k_{\rm i}h_{\rm i}}{k_{\rm i}\tau_{\rm w}}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Definitions of dimensionless variables.