## 1 Background Study
However, the equation built by us did not require Laplace Transformation or inversion but simple leveraging on a single equation that can be used to compute pressure change with respect to the location and time without restriction of the type of flow. The origin of the fractured model study in well testing begun in 1960 when a study was conducted on the seepage of reservoir fluid along the fractured strata from which the term dual porosity was derived [14]. The work on the dual porosity model continued in 1963 when an ideal homogeneous model was built that comprised of two components, the homogeneous reservoir model and the network of fractures. In reservoir simulation, to describe a hydraulic fractured well, a dual porosity grid that contains the well and the hydraulically fractured area is used because an infinite conductive grid with small volume is not easy to be actualized by a normal single porosity. The introduction of terms such as storage coefficient and inter-porosity coefficient took place, which governed the fluid flow between the dual porosity medium, thus establishing a solid foundation for the dual porosity model. Our main contribution is to define a new equation that has universal application to any flow regimes. This means that, unlike other studies that generated several equations specific for flow through different mediums and different regimes, we managed to construct a single equation that works universally in all of the flow regimes that will occur for a well that is hydraulically fractured. This means that the inclusion of the dual porosity element can be eliminated using the equation that we built. Figure 1 shows dual porosity model from a simulator where the darker line indicated fractures originated from hydraulic fractures. Figure 2 shows the early time pressure response around hydraulically fractured well. The intensity on Figure 2 indicates the magnitude of pressure whereby around the hydraulically fractured well, a stark contrast can be seen. These outcomes take place because the flow is dominant from fracture to matrix instead of matrix to matrix. Thus, the pressure drops is more intense across the fracture. Figure 1: Dual porosity Assigned Grid (Dark blue) for Hydraulic Fracture. The equation of shale matrix and fracture networks were coupled and an equation was generated. The source function used in this study was [PERSON]'s function, which was modified to be used in Laplace Transformation [13]. \[\overline{P}_{D}=\overline{q}_{fd}K_{o}\bigg{(}\sqrt{x_{D}^{2}+y_{D}^{2}}\sqrt{ f(s)}\bigg{)} \tag{2}\]  
Equation (2). Shale Matrix and Hydraulic Fracture Coupling. \[f(s)=s\frac{\omega S(1-\omega)+\lambda}{S(1-\omega)+\lambda} \tag{3}\]  
Equation (3). Bessel Function with Laplace Transformation (Source Function). The final equation used in this study is the superposition equation as expressed in  
\[\overline{P}_{FDi}=\sum_{k=1}^{n_{p}}\int_{y_{wDL,\mu}}^{y_{wDL,\mu}}\overline{ q}_{FDK}\bigg{[}f(s)\sqrt{\big{(}x_{Di}-x_{wD}\big{)}^{2}+\big{(}y_{Di}-u \big{)}^{2}}\bigg{]}du \tag{4}\]  
Equation (4). Superposition Equation. This equation can be used to compute and model the pressure in both the shale matrix and in fracture networks, which actually caters for the linear flow during the flow into the fracture. However, the equation constructed cannot be used to model the subsequent flows that occurred after the linear flow such as the elliptical flow (around the well and fracture networks) and the radial flow (reservoir boundary dominated flow). In order to simplify the case, we managed to build an equation that can govern all of the flow regimes, and the equation that we built does not require Laplace transformation in order to be used. Another study conducted was on constructing a semi-analytical model for pressure transient analysis of a hydraulically fractured well with fracture orientation in an isotropic reservoir. A set of diffusivity equations was developed using a nodal analysis technique, which describes the flow in the hydraulic fracture.