## 3 Validation
### Parameters and Details
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Author** & **Model** & **Annotation** \\ \hline Carman-Kozeny & \(\frac{k}{\sigma}=\frac{1}{\mu_{0}}\frac{\varphi^{3}}{(1-\varphi)^{2}}\) & \(\kappa_{0}\) is the Kozeny constant, taking a value between 4.17 and 5 [18] \\ Gebart & \(\frac{k}{\sigma}=C\Big{(}\sqrt{\frac{1-\varphi_{c}}{1-\varphi}}-1\Big{)}^{5/2}\) & \(C\) is a geometric factor, \(C=16/9\sqrt{2}\pi\) and \(\varphi_{c}=1-\pi/4\) for a square array \\ [PERSON] & \(\frac{k}{\sigma}=0.117\varphi^{4/57}\) & / \\ (1997) [36] & \(\frac{k}{\sigma}=0.117\varphi^{4/57}\) & / \\ [PERSON] et al. & \(\frac{k}{\sigma}=\frac{5.55}{\alpha\ln(1-\varphi)-1}\) & / \\ [PERSON] et al. & \(\frac{k}{\sigma}=b_{1}\Big{(}\sqrt{\frac{1-\varphi_{c}}{1-\varphi}}-1\Big{)}^{ 2}\cdot e^{b_{1}(1-\varphi)}\) & \(b_{1}\) and \(b_{2}\) are curve-fitting parameters. \\ [PERSON] et al. & \(\frac{k}{\sigma}=b_{1}\Big{(}\sqrt{\frac{1-\varphi_{c}}{1-\varphi}}-1\Big{)}^{ 2}\cdot e^{b_{2}(1-\varphi)}\) & For example, \(b_{1}=17.01\) and \\ (2009) [21] & \(\frac{k}{\sigma}=C_{1}\Big{(}\sqrt{\frac{1-\varphi_{c}}{1-\varphi}}-1\Big{)}^{ C_{2}}\) & \(C_{1}=0.491,C_{2}=2.31\) \\ [PERSON] & \(\frac{k}{\sigma}=0.204\varphi\) [45] & / \\ [PERSON] and [PERSON] & \(\frac{k}{\sigma}=b_{1}\Big{(}\sqrt{\frac{1-\varphi_{c}}{1-\varphi}}-1\Big{)}^{ \alpha}\cdot e^{b_{2}(1-\varphi)}\) & For example, \(n=1.194,b_{1}=8.664\), and \(b_{2}=-2.229\times 10^{-14}\) for the sandstone \\ [PERSON] and [PERSON] (2022) [13] & \(\frac{k}{D_{e}^{2}}=\alpha e^{\beta\varphi}\Big{(}1-e^{-s_{p}/s_{f}}\Big{)}\) & \(\begin{array}{c}\alpha=9.18\times 10^{-5}\Big{(}\frac{s_{p}}{s_{f}}\Big{)}^{-1.35}\\ \beta=7.39\Big{(}\frac{s_{p}}{s_{f}}\Big{)}^{0.40422}\end{array}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main permeability models. large-sized proppant, [PERSON]'s model (2022) [13] is adopted. These two models are suitable for the small-sized and large-sized cases, respectively. Another important problem is how to calculate reservoir permeability from fracture permeability. Generally, the greater the fracture density, the better the reservoir permeability. This paper focuses on the influence of proppant size on the reservoir pressure field and productivity. Therefore, it assumes that the crack density is a fixed value for different proppant types. Therefore, if the fracture permeability is increased by n times due to the change in proppant size, it can be considered that the overall reservoir permeability is also increased by n times. In this study, the large size and small size are relative. Small size mainly refers to the fact that the size of the proppant is much smaller than the crack opening; that is, the proppant exists in the crack in the form of a continuous package. A large-size proppant means that the proppant size is equivalent to the crack opening, and the proppant exists in the crack in the form of dispersion. Therefore, large size and small size are concepts relative to the crack opening. For example, the proppant with a diameter of 1 mm can be considered a small-size proppant (assuming a crack opening of 10 mm), and the proppant with a diameter of 1 mm can also be considered a large-size proppant (assuming a crack opening of 1 mm). Therefore, the large size and small size are relative concepts. In the following discussion, it is assumed that the crack opening is 1 mm, so the proppant of 0.1 mm can be regarded as the small-size proppant, and the proppant of 1 mm can be regarded as the large-size proppant.