## Appendix B Derivation of the Solution  
Firstly, the Laplace transforms with respect to \(t_{\text{D}}\) for dimensionless fracture and matrix pseudopressure are defined as following, respectively:  
\[\overline{p}_{fD}(r_{\text{D}},z_{D},s) =\int\limits_{0}^{+\infty}e^{-st_{D}}p_{fD}(r_{D},z_{D},t_{D})dt_{D} \tag{20}\] \[\overline{p}_{m1D}(r_{D},z_{D},s) =\int\limits_{0}^{+\infty}e^{-st_{D}}p_{m1D}(r_{D},z_{D},t_{D})dt _{D}\] (21) \[\overline{p}_{m2D}(r_{D},z_{D},s) =\int\limits_{0}^{+\infty}e^{-st_{D}}p_{m2D}(r_{D},z_{D},t_{D})dt _{D} \tag{22}\]  
Based on the Laplace transforms, the control equations in Laplace space are:  
\[\frac{1}{r_{D}}\frac{\partial}{\partial r_{D}}\bigg{(}r_{D}\frac{\partial \overline{p}_{fD}}{\partial r_{D}}\bigg{)}+\frac{1}{h_{1D}^{2}}\frac{\partial }{\partial z_{D}}\bigg{(}\frac{\partial\overline{p}_{fD}}{\partial z_{D}} \bigg{)}=sf(s)\overline{p}_{fD} \tag{23}\]  
\[\frac{1}{r_{D}}\frac{\partial}{\partial r_{D}}\bigg{(}r_{D}\frac{\partial \overline{p}_{m2D}}{\partial r_{D}}\bigg{)}+\frac{1}{h_{2D}^{2}}\frac{ \partial}{\partial z_{D}}\bigg{(}\frac{\partial\overline{p}_{m2D}}{\partial z _{D}}\bigg{)}=\frac{1}{\sigma}\widetilde{sp}_{m2D} \tag{24}\]  
where, \(h_{1D}=h_{D}\sqrt{\kappa_{1}}\), \(h_{2D}=h_{D}\sqrt{\kappa_{2}}\), \(f(s)=\frac{\lambda+\sqrt{\kappa_{1}}(1-\omega)s}{\lambda+\sqrt{\kappa_{1}}(1- \omega)s}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Dimensionless Parameters** & **Expression** & **Dimensionless Parameters** & **Expression** \\ \hline Dimensionless radius & \(r_{D}=\frac{r}{\tau_{\infty}}\) & Dimensionless time & \(t_{D}=\frac{\alpha k_{\text{D}}\mu}{\mu_{p}\left[(\phi_{\text{c}})_{m}+(\phi_{ \text{c}})_{\mu}\right]}\) \\ Dimension inner region radius & \(R_{1D}=\frac{R_{1}}{\tau_{\infty}}\) & Dimensionless natural fracture & \(p_{fD}=\frac{k_{D}\mu(p_{\text{c}}-p_{fD})}{\alpha\mu_{p}\mu_{p}\mu_{p}}\) \\ Dimensionless reservoir & \(R_{cD}=\frac{R_{1}}{\tau_{\infty}}\) & Dimensionless matrix & \(p_{m1D}=\frac{k_{D}\mu(p_{\text{c}}-p_{fD})}{\alpha\mu_{p}\mu_{p}\mu_{p}}\) \\ Dimensionless vertical & \(z_{D}=\frac{z}{h}\) & Dimensionless matrix & \(p_{m2D}=\frac{k_{D}\mu(p_{\text{c}}-p_{fD})}{\alpha\mu_{p}\mu_{p}\mu_{p}}\) \\ Dimensionless vertical & \(z_{nD}=\frac{z_{n}}{h}\) & Dimensionless production rate & \(q_{nD}=\frac{q_{n}(t)\alpha\mu_{p}\mu_{p}\mu_{p}}{k_{D}\mu(p_{\text{c}}-p_{fD })\mu(p_{\text{c}}-p_{fD})}\) \\ Dimensionless reservoir & \(h_{D}=\frac{h}{\tau_{\infty}}\) & Storativity ratio of natural fracture, \(\omega\) & \(\omega=\frac{(\phi_{\text{c}})_{\mu}}{(\phi_{\text{c}})_{\mu+\text{c}}}\) \\ Dimensionless perforation & \(h_{nD}=\frac{h_{n}}{h}\) & Interporosity flow coefficient & \(\lambda=\alpha\frac{K_{\text{c}}}{K_{\text{c}}^{2}}\frac{r_{\text{c}}^{2}}{ \left(\frac{\lambda}{\left(\phi_{\text{c}}\right)}\right)_{nD}}\) \\ Permeability anisotropy factor & \(\kappa_{1}=\frac{k_{D}}{\xi_{\mu}}\) & Diffusivity ration between outer and inner regions, \(\delta\) & \(\sigma=\frac{k_{D}\omega_{\text{c}}}{(\phi_{\text{c}})_{\mu_{\text{c}}}}/ \frac{k_{D}}{(\phi_{\text{c}})_{\mu_{\text{c}}}\mu_{p}}\) \\ Permeability anisotropy factor & \(\kappa_{2}=\frac{k_{D}\omega_{\text{c}}}{k_{\text{c}}}\) & Mobility ratio between outer and inner regions, \(M\) & \(M=\frac{k_{D}\omega_{\text{c}}}{(\phi_{\text{c}})_{\mu_{\text{c}}}\mu_{p}}/ \frac{k_{D}}{(\phi_{\text{c}})_{\mu_{\text{c}}}\mu_{p}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 10: The definitions of dimensionless parameters.