# Coupled Numerical Method for Modeling Propped Fracture Behavior
## Materials and Methods
### Model Description
#### 2.1.2 Reservoir and Proppant  
As described in the introduction, the fracture conductivity is investigated after hydraulic fractures are created, propped, and the high-pressure hydraulic pumps placed at the surface are stopped, entailing the formation to close again. Boundary conditions were defined for the reservoir and the proppant itself to investigate the phenomenon relevantly. The hydrocarbon-bearing reservoir characteristics directly impact the formation Young's modulus and Poisson's ratio, closure pressure, fluid pressure, and fluid flow. Therefore, the boundary condition for rock Young's moduli was defined for a wide range providing an environment to investigate the effect of formation elasticity to proppant embedment and fracture conductivity in soft, well-consolidated, or even tough deposits, igneous or metamorphic rocks. The minimum value for formation Young's moduli was defined as 1 GPa according to the research conducted by [PERSON] et al. [7], who examined proppant embedment and fracture conductivity in soft formations. Max and mean values were defined as 30 and 15 GPa referred to [PERSON] and [PERSON]'s scientific work [29] in which more than 500 core samples have been measured to determine rock Young's moduli. Since Poisson's ratio of the reservoir does not have a significant influence on the examined phenomenon, it was fixed to 0.22, referring to the article published by [PERSON] et al. [30]. The latter studied the depth of proppant embedment in hydraulic fractures. For the relation of overburden, pore, and closure pressure the widely known Hubbert and Willis [31] formula was used:  
\[\sigma_{c}=\
u/(1-\
u)\ ^{*}\left(\sigma_{v}-ap_{r}\right)+ap_{r}, \tag{1}\]  
where \(\sigma_{c}\) is closure stress / pressure [MPa], \(\
u\) is [PERSON]'s ratio [-], \(\sigma_{v}\) is overburden or vertical stress [MPa], \(\alpha\) is [PERSON]'s poroelastic constant [-] and \(p_{r}\) is pore pressure [MPa]. One should note that this equation proposes that the in situ horizontal stress is generated by the uniaxial vertical strain without any tectonic activity. The pore pressure in the reservoir is supposed to be hydrostatic, while the overburden pressure was considered by  
Figure 2: Model geometry. the weight of the rocks assuming a medium sandstone density of 2.74 g/cm\({}^{3}\): according to [PERSON] et al. [32]. They dealt with rock moduli and density determination using wireline data. Finally, the limits of closure pressure values used in our research were defined as 2139 and 13,904 psi that equals \(\sim\)14.7 and 95.9 MPa in SI units, respectively. These closure pressure values also cover a wide range for investigating the effect of fracture pressure in shallow, medium, and even deep wells with a high amount of overburden. Another relevant parameter to investigate the fracture behavior and fracture conductivity is the fluid flow in the fracture. In the case of hydraulic fracturing, several flow regimes are interpreted during the production period of the well. [PERSON] et al. [33] defined four flow patterns for hydraulically fractured wells: fracture linear flow, bilinear flow, formation linear flow, and pseudo-radial flow. Figure 3 represents the schematic of them. The conductivity we investigated is independent of the flow patterns and can be used for all the cases. According to the research published by [PERSON] et al. [34], inertial flow is one of the parameters that can reduce the effective fracture permeability and conductivity significantly. The flow rate in our model was determined in a way that allows one to consider pressure drop due to not only Darcy but also non-Darcy effects, enabling one to conclude a real fractured well. Inertial forces occur during fluid flow at high velocities; therefore, conductivity was investigated at 0.144 kg/s mass flow, meaning a water flow rate of 78.4 BPD.