## Appendix A Symbol Description  
_Nomenclature_  
\(C\) & wellbore storage coefficient, m\({}^{3}\)/Pa  
\(C_{\text{fD}}\) & fracture conductivity coefficient  
\(c_{\text{n}}\) & total compressibility of hydraulic fracture, Pa\({}^{-1}\)  
\(c_{\text{t}}\) & total compressibility of gas reservoir, Pa\({}^{-1}\)  
\(h\) & reservoir thickness, m  
\(k\) & reservoir permeability, m\({}^{2}\)  
\(k_{\text{fi}}\) & initial fracture permeability, m\({}^{2}\)  
\(k_{\text{i}}\) & initial reservoir permeability, m\({}^{2}\)  
\(L_{\text{fj}}\) & length of the \(j\)th hydraulic fracture, m  
\(n\) & hydraulic-fracture number  
\(p\) & reservoir pressure, Pa  
\(p_{\text{f}}\) & fracture pressure, Pa  
\(p_{\text{i}}\) & initial reservoir pressure, Pa  
\(p_{\text{sc}}\) & pressure at standard condition, Pa  
\(q\) & flow rate of point source, m\({}^{3}\)/s  
\(q_{\text{f}}\) & flow-rate density, m\({}^{2}\)/s  
\(Q_{i}\) & production rate of the \(i\)th hydraulic fracture, m\({}^{3}\)/s  
\(Q_{\text{sc}}\) & well production rate, m\({}^{3}\)/s  
\(r\) & radial distance, \(r=\sqrt{x^{2}+y^{2}}\), m  
\(r_{\text{w}}\) & wellbore radius, m  
\(s\) & Laplace transform variable  
\(S_{\text{f}}\) & skin factor  
\(t\) & time, s  
\(T\) & reservoir temperature, K  
\(T_{\text{sc}}\) & temperature at standard condition, K  
\(w_{\text{f}}\) & fracture width, m  
\(x_{i},y_{i}\) & \(x_{i}\)- and \(y_{i}\)-coordinates, m  
\(Z\) & Z-factor of gas  
\(\mu\) & gas viscosity, Pa\(\cdot\) s  
\(\gamma\) & permeability modulus, \((\text{Pa}\cdot\text{s})/\text{Pa}^{2}\)  
\(\psi\) & reservoir pseudo-pressure, Pa\({}^{2}\)/\((\text{Pa}\cdot\text{s})\)  
\(\psi_{\text{f}}\) & fracture pseudo-pressure, Pa\({}^{2}\)/\((\text{Pa}\cdot\text{s})\)  
\(\psi_{\text{i}}\) & initial reservoir pseudo-pressure, Pa\({}^{2}\)/\((\text{Pa}\cdot\text{s})\)
Appendix B Pressure Distribution in a Gas Reservoir with a Vertical Fractured Well with Multiple Hydraulic Fractures Producing at a Constant Rate  
With the aid of Equation (11), Equations (7)-(10) can be rewritten as  
\[\frac{\partial^{2}\xi_{\rm D}}{\partial r_{\rm D}^{2}}+\frac{1}{r_{\rm D}}\frac{ \partial\xi_{\rm D}}{\partial r_{\rm D}}=\frac{1}{1-\gamma_{\rm D}\xi_{\rm D}} \frac{\partial\xi_{\rm D}}{\partial t_{\rm D}}, \tag{11}\]  
\[\xi_{\rm D}(r_{\rm D},t_{\rm D}=0)=0, \tag{12}\]  
\[\xi_{\rm D}(r_{\rm D},t_{\rm D}=0)=0, \tag{13}\]  
\[\lim_{r_{\rm D}\to 0}r_{\rm D}\frac{\partial\xi_{\rm D}}{\partial r_{\rm D}}=-q_{ \rm D}(r_{\rm D}=0,t_{\rm D}). \tag{14}\]  
The perturbation technique of \(\gamma_{\rm D}\) was employed to obtain that  
\[\xi_{\rm D}=\xi_{\rm D0}+\gamma_{\rm D}\xi_{\rm D1}+\gamma_{\rm D}^{2}\xi_{\rm D 2}+\cdots, \tag{15}\]  
\[-\frac{1}{\gamma_{\rm D}}\ln[1-\gamma_{\rm D}\xi_{\rm D}]=\xi_{\rm D}+\frac{1 }{2}\gamma_{\rm D}\xi_{\rm D}^{2}+\cdots, \tag{16}\]  
\[\frac{1}{1-\gamma_{\rm D}\xi_{\rm D}}=1+\gamma_{\rm D}\xi_{\rm D}+\gamma_{\rm D }^{2}\xi_{\rm D}^{2}+\cdots. \tag{17}\]  
Considering a small value of \(\gamma_{\rm D}\) in general, the zero-order perturbation solution was accurate enough for engineering requirements. So, Equations (11)-(14) can be rewritten as  
\[\frac{\partial^{2}\xi_{\rm D0}}{\partial r_{\rm D}^{2}}+\frac{1}{r_{\rm D}} \frac{\partial\xi_{\rm D0}}{\partial r_{\rm D}}=\frac{\partial\xi_{\rm D0}}{ \partial t_{\rm D}}, \tag{18}\]  
\[\xi_{\rm D0}(r_{\rm D},t_{\rm D}=0)=0, \tag{19}\]  
\[\xi_{\rm D0}(r_{\rm D}\rightarrow\infty,t_{\rm D})=0, \tag{20}\]  
\[\xi_{\rm D}(r_{\rm D},t_{\rm D}=0)=0, \tag{12}\]0  
Laplace transform was employed to solve the above model and it is defined by  
\[\xi_{\rm D}(r_{\rm D},t_{\rm D}=0)=0, \tag{12}\]1  
where \(f\) is an arbitrary variable in real space.