## 3 Results and Discussion
### Discrete Solution  
Equation (17): Pressure Drop Solution at Wellbore. \[ \begin{split} p_{flowing}(r=r_{w},t)=p_{i}-\frac{70.6B\mu}{kh} \frac{r_{w}q}{x_{f}}\bigg{[}-Ei\left(-\frac{948.1\phi\mu c_{t}r_{w}^{2}}{kt} \right)\bigg{]}-\\ \lim_{n\rightarrow\infty}\sum_{j=1}^{n}\frac{70.6B\mu}{kh}\frac{ q}{x_{f}}\left[-Ei\left(-\frac{948.1\phi\mu c_{t}\Big{(}r_{w}+\frac{x_{f}-r_{w}}{n}j \Big{)}^{2}}{kt}\right)\right]\frac{x_{f}-r_{w}}{n}\end{split}  \tag{17}\]  
The developed equation showed a perfect superimposition. The developed analytical model managed to provide a smooth computation of all of the three phases, linear, transition and pseudo-radial, using a single equation. Moreover, the final stage is real pseudo-radial flow, not conventional radial. The developed analytical model matched with the numerical calculation. Thus, the analytical model again managed to depict all of the stages of flow using a single relatively easy inversible equation, which is unlike the numerical solution. All of the three stages of flow, the linear, transition and pseudo-radial flow schematics are also shown in Figures 9-11, respectively. In a study conducted on pressure transient analysis in an elongated linear flow system, the applicability of the linear equation for interference, drawdown and build-up tests was thoroughly studied. In this study, an equation was built and the equation was utilized for gas, oil and steam geothermal wells. The built equation is shown in Equation (14). Figure 11: Pseudo-Radial Flow in Late time. Figure 9: Early Time Linear Flow. However, the drawback of this equation is that there is no consideration of pseudo-radial flow in a later time. Although the surface area at the tip is less, there is still flow into the section, which is not captured by the study. However, the newly developed equation by the author accounts for the flow from both tips of the fracture [19]. This can be seen in Figures 12 and 13, which clearly demonstrate the difference between the outcomes of the two studies. The flow into the tip is the reason why, eventually, the flow turns elliptical. In the previous study, there was no consideration of the pseudo-radial flow and the transition period [19]. \[\textbf{D = 0.00026367}\frac{\textbf{k}}{\phi\mu c_{t}} \tag{18}\] \[p =p_{i}-\frac{141.2 qB\mu}{khx_{f}}\left[\sqrt{\pi 0.0002637}\frac{ kt}{\phi\mu c_{t}}-\pi x\right]\]  
Equation (18): Elongated Linear Flow Equation. Figure 12: Pressure Iso-line (Conventional Analytical for early time linear flow Model). Figure 13: Pressure Iso-line Assumption in Developed Analytical Model for early time semi-linear flow. Another similar study was conducted focused on building an analytical solution for transient flow in a hydraulically vertically fractured well in an elliptical shaped and an isotropic reservoir producing at a constant rate and pressure. An elliptical model was suggested because the radial model is only applicable for a homogeneous and isotropic reservoir. There were a few underlying assumptions used in developing the equation for the transient model linear flow such as it being an isotropic, homogeneous, horizontal, infinite reservoir with constant thickness, permeability and porosity. The reservoir fluid was assumed to be slightly compressible with constant viscosity. All of the properties were assumed to be independent of pressure. Laminar flow was considered with negligible gravity effect [20]. The analytical model built via this study is an approximated model unlike the model built by the author. The model built by the author of this study is an exact analytical solution for this behavior. On top of that, the mode built in the previous study is an equation for transient flow only, not pseudo-radial and linear. In order to cater for elliptical flow, a different permeability in the x- and y-direction was introduced, which brings in the term permeability anisotropy [20].