## 1 Background Study
The coupling of the source solution and discrete fracture solution generated the semi-analytical solution. The generated semi-analytical solution was then validated using the numerical solution [15]. There were two equations generated in the paper which were for the flow from one fracture node to another and the flow from the fracture to the vertical well, as shown in Equations (5) and (6). \[\widetilde{p}_{fDk+1}-\widetilde{p}_{fDk}=\frac{2\pi}{F_{cD,i}}\int_{l_{Dk}}^{ l_{Dk+1}}\Big{[}\widetilde{q}_{fwDk}+\widetilde{q}_{fDk}(l_{D}-l_{Dk})\Big{]}dl_{D} \tag{5}\]  
Equation (5). Flow between Fracture Nodes. \[\widetilde{p}_{Dk,i}(x_{Dk}y_{Dk},x_{Di}y_{Di},s)\ =\ \sum_{k=1}^{N_{l}+1}\sum_{i= 1}^{N_{l}}\widetilde{q}_{Dj}R_{Dk,i}(x_{Dk},y_{Dk},x_{Di},y_{Di},s) \tag{6}\]  
Figure 2: Early Time Pressure around Hydraulic Fractured Well. Equation (6). Flow from Fracture to Vertical Well. In this study, the equation is discretized where the pressure drop is computed by the fluid flow from every interconnecting fracture and matrix surface, as shown in Equation (6). This approach differed from our work where a single equation was constructed to compute the pressure response in the linear flow regime analytically. The analytical approach always yields a more accurate solution than any numerical approach; thus, our work will deliver a more accurate result. The boundary condition in this study and our study was treated differently. In this study, since the flow rate along the fracture plane was added individually, the sum that was used to compute the total pressure drop indicated that the pressure drop was considered uniform instead of a uniform flow assumption. Another study was conducted on building semi-analytical solutions for multi-fractured horizontal wells in box-shaped reservoirs. In this study, Laplace and Fourier Transforms were used to generate the semi-analytical solution. Prior to undergoing the transformations, two main models were predefined such as the reservoir and fracture model, which were used to represent the models, respectively. In this study, the semi-analytical solution yielded a higher flowrate in the outermost part of the fracture, which resembles the assumption of a constant pressure drop over the fracture interval. This underlying assumption was different from our work, which assumed a uniform flux into the fracture, thus creating a gap in the literature. In this study, there were six flow regimes established altogether, namely bilinear flow, first linear flow, first radial flow, second linear flow, second radial flow and, finally, boundaries dominated flow. This finding contradicted the common flow regimes or patterns often established in a fractured reservoir, which begins with a linear flow from the matrix to the fracture, a pseudo-radial flow that is established as the flow from the matrix enters the side and at the tip of the fractures creating an elliptical behavior, and, finally, the radial flow, which is a boundary dominated flow that is established at late stage [16]. Basically, we defined three different flow regimes, unlike this study, [6]. The main contributor to the difference between our study and the author study is the orientation of the well. In our study, all of the well was vertical with a single hydraulic fracture, while in the literature study, the well was horizontal and had multiple fractures. Apart from these differences, the flow nature was of an infinite conductivity in the literature, while in our study it was a uniform flux. In a study conducted on transient pressure behavior for a well with a finite conductivity vertical fracture, a mathematical model was developed. There were a few underlying assumptions used to develop the model. The first assumption was that it was an isotropic, homogeneous, horizontal, infinite and slab reservoir. The fluid in the reservoir was slightly compressible, which means the viscosity and compressibility were both constant. Fluid entered the wellbore only from the fracture, which neglected the direct flow from the formation to the wellbore.