## 4 Hydraulic Fracturing Model
### Unconventional Fracture Model (UFM)  
The fracturing model applied in this study is UFM, which simulates vertical fracture propagation, rock deformation, and fluid flow in the complex fracture network created during a fracture treatment. The basic governing equation describes the fluid flow in the fracture network, the fracture deformation, and the fracture propagation/interaction criterion [21]. The Equations (8) and (9) are the continuity equations assuming that fluid flow propagates along fracture network with mass conservation,  
\[\frac{\partial q}{\partial s}+\frac{\partial\Big{(}H_{fl}\overline{w}\Big{)} }{\partial t}+q_{L}=0 \tag{8}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Stage\#**} & \multicolumn{2}{c}{**Average Rate**} & \multicolumn{1}{c}{**Propants**} & \multicolumn{1}{c}{**Propant**} & \multicolumn{1}{c}{**Clean Volume**} \\  & **(m\({}^{3}\)/min)** & **(kg)** & **Concentration (kg/m\({}^{3}\))** & **(m\({}^{3}\))** \\ \hline \multirow{8}{*}{1} & Break Down Linear Gel & 1.6 & 0 & 0 & 5.7 \\  & Acid & 2.4 & 0 & 0 & 3.8 \\  & Pad 30 \# Borate XL & 4.8 & 0 & 0 & 50.6 \\  & 0.25 \# 16/30 White & 4.8 & 454 & 30.1 & 15.1 \\  & Pad 30 \# Borate XL & 4.8 & 0 & 0 & 94.1 \\  & 1.0 \# 16/30 White & 4.8 & 8744 & 119.8 & 73.0 \\  & 2.0 \# 16/30 White & 4.8 & 13,907 & 239.4 & 58.1 \\  & 3.0 \# 16/30 White & 4.8 & 17,603 & 359.2 & 49.0 \\  & 4.0 \# 16/30 White & 4.8 & 25,223 & 478.6 & 52.7 \\  & 4.0 \# 16/30 Fiber Prop & 4.8 & 4953 & 476.3 & 10.4 \\  & Flush 30 \# Linear Gel & 4.9 & 0 & 0 & 54.8 \\ \hline \multirow{8}{*}{3} & Break Down Linear Gel & 1.6 & 0 & 0 & 3.2 \\  & Pad 30 \# Borate XL & 5.1 & 0 & 0 & 33.1 \\  & 0.25 \# 16/30 White & 5.1 & 461 & 30.1 & 15.4 \\  & Pad 30 \# Borate XL & 5.1 & 0 & 0 & 76.1 \\ \cline{1-1}  & 1.0 \# 16/30 White & 5.1 & 8420 & 119.8 & 73.5 \\ \cline{1-1}  & 2.0 \# 16/30 White & 5.1 & 13,697 & 239.4 & 57.1 \\ \cline{1-1}  & 3.0 \# 16/30 White & 5.1 & 18,067 & 359.2 & 50.2 \\ \cline{1-1}  & 4.0 \# 16/30 White & 5.1 & 22,263 & 478.6 & 46.4 \\ \cline{1-1}  & 4.0 \# 16/30 Fiber Prop & 5.1 & 5420 & 476.3 & 11.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stage #1 pumping design for hydraulic fracturing treatment. Figure 4: Completion design of SANDY03. \[q_{L}=2h_{L}u_{L} \tag{9}\]  
where \(q\) is the local flow rate inside the hydraulic fracture along the length, \(\overline{w}\) is an average width, \(H_{fl}\) is the local height of the fracture occupied by fluid, and \(q_{L}\) is the leak-off volume rate through the wall of the hydraulic fracture into the rock matrix per unit length. Through the [PERSON]'s leak-off model, \(q_{L}\) is yielded by \(h_{L}\), leak-off height, times velocity, \(u_{L}\), at which fracturing fluid infiltrates into a surrounding permeable medium. For a vertical fracture (Figure 5) in a layered formation with variable \(\sigma_{\rm hmin}\) and fluid pressure \(p\), the width profile can be determined from an analytical solution given as  
\[w(x,\,y,\,z)=w(p(x,\,y),\,h,\,z) \tag{10}\]  
and the varying parameter, height of the fracture \(h\), is calculated based on the approach described in [PERSON] et al. (2011). The fracture height is calculated based on the pressure at each location of the fracture by matching \"stress intensity factors, \(K_{lu}\) and \(K_{ll}\)\" as in Equations (11) and (12).