## 1 Background Study
There were two equations generated in this study for the fracture and reservoir models. There were 6 flow regimes mentioned in this study as shown in Figure 3. However, these flow regimes only exist for horizontal wells. In this study, it was mentioned that infinite conductivity does not work in all cases where, for some cases such as for large or very low-capacity fractures, finite conductivity or uniform flow should be used instead. The author of this study mainly wanted to produce a general solution for the pressure transient of a well intersecting a vertical fracture with finite conductivity. The following equations were derived to be used as the solution [17]. Despite a similarity between the study cited here and our work, the cited work experienced some setbacks in matching extreme cases where the conductivity was extremely low or high. This probably happened because the transient equation was used generally for all of the flow regimes where the inaccuracies may have taken place once the radius of investigation had reached the reservoir boundary. \[\frac{\partial^{2}P_{f}}{\partial x^{2}}+\frac{u}{k_{f}}\frac{q_{f}(x,t)}{wh} =\frac{\phi_{f}\mu c_{ft}}{k_{f}}\frac{\partial p_{f}}{\partial t} \tag{7}\]Equation (7). Partial Differential Equation for Fracture (Unsteady State). \[ \begin{split} P_{fD}(x_{D},t_{D})&=\frac{x_{f}}{w} \sqrt{\frac{k\varphi c_{t}\tau}{k_{f}\phi_{f}c_{ft}}}\sum_{n=-}^{\infty}\int_{0}^ {t_{D}}\Bigg{\{}\frac{e^{-\left[\frac{(x_{D}-x)^{2}}{4\tau\left(k_{f}\phi c_{t} /\lambda_{f}c_{ft}\right)}\right]}}{\sqrt{\tau}}\\ &\quad-\int_{2n-1}^{2n+1}q_{fD}(x,\tau)\frac{e^{-\left[\frac{(x_ {D}-x)^{2}}{4\tau\left(k_{f}\phi c_{t}/\lambda_{f}c_{ft}\right)}\right]}}{2 \sqrt{t_{D}-\tau}}dx^{\prime}\Bigg{\}}d\tau\end{split}  \tag{8}\]  
Equation (8). Fracture Model Solution for Finite Conductivity. The developed solution for the fracture model in a finite conductivity case only works for the linear flow that happens as the fluid from the matrix starts flowing into the fracture where the previous fracture dominated the flow or the post linear flows cannot be modeled by the solution. This rendered the equation totally inaccurate because the main flow occurred from the fracture into the well, which was not modelled in the literature. In our work, not only did we model this flow but also any subsequent flows that would take place in a hydraulically fractured vertical well. We managed to produce a single equation that provided the solution to all of the comprised stages in fractured well testing. An analytical model was developed for pressure transient analysis for multi-fractured horizontal wells in heterogenous shale reservoirs. This analytical model catered for many assumptions, which were neglected in many established models, such as reservoir heterogeneity, gas slippage, adsorption/desorption and diffusion effect. The model was divided into four main divisions, namely upper/lower regions, outer-reservoir regions, inner reservoir regions and fracture regions. The model created in this study is different from the author's model, which has a single equation coupling the fracture and reservoir models. In their model, all of the flow regimes have their own equations, which denied the general application of a single equation to all cases, as in our study [18]. Figure 3: 6 Flow Regimes for a Multi-fractured Horizontal Well. The following Equation (9) defined the fracture length segmentation  
\[ \begin{array}{c}\Delta L=\frac{x_{f}-r_{w}}{n}\\ L_{n}=r_{w}+n\Delta L\\ f(L_{n})=f(r_{w}+n\Delta L)\end{array}  \tag{9}\]  
Equation (9): Fracture Length Segmentation. The flow rate per unit area \(u(L)\) is defined by Equation (10). \[2\int_{0}^{x_{f}}u(L)dL=q \tag{10}\]  
Equation (10): Flow Rate per Unit Area  
When the flowrate is uniform, Equation (10) is modified to Equation (11). \[ \begin{array}{c}u(L)=\frac{q}{2x_{f}}\\ 2\int_{r_{w}}^{x_{f}}\frac{q}{2x_{f}}dL=\frac{x_{f}-r_{w}}{x_{f}}q\\ 2\int_{0}^{r_{w}}\frac{q}{2x_{f}}dL=\frac{r_{w}}{x_{f}}q\end{array}  \tag{11}\]  
Equation (11): Flow Rate per Unit Area (Uniform).