## 5 Conclusions  
This paper analyzes the influence of two typical proppant types on fracture and reservoir permeability and then discusses the differences in reservoir pressure field and production rate between reservoir 1, containing small-sized proppant, and reservoir 2, containing large-sized proppant. The main findings are concluded as follows.  
(1) On the premise of the same porosity, the permeability of the fracturing reservoir 1 containing large proppant is higher, which makes the average pressure of the reservoir drop faster, and the natural gas production efficiency is higher. Taking a single well as an example, the natural gas production efficiency is increased by 19.24 times (from 5.546 m\({}^{3}\)/d to 106.7 m\({}^{3}\)/d) at the initial stage of production (\(t=1\) d) after the use of large-sized proppant. In the later period of exploitation (\(t=3000\) d), the natural gas exploitation efficiency is increased by 17.01 times (from 3.037 m\({}^{3}\)/d to 51.65 m\({}^{3}\)/d) when adopting the large-size proppant.  
Figure 24: Permeability magnification of fracture when increasing the proppant size under the condition of same porosity. \(d\) is the equivalent diameter of the small-size proppant and \(D\) is the equivalent diameter of the large-sized proppant. The upper part is from Gillard.  
(2) Under the same conditions, increasing the number of production wells can significantly accelerate the reduction of reservoir pressure and significantly improve the efficiency of natural gas production. However, the increased times of the production efficiency will gradually decrease with time. Taking the reservoir containing large-size proppant as an example, the production efficiency of natural gas will increase by four times (from 106.7 m\({}^{3}\)/d to 426.7 m\({}^{3}\)/d) at the initial stage of production (\(t\) = 1 d). In the later period of exploitation (\(t\) = 3000 d), the natural gas exploitation efficiency is increased by 2.12 times (from 51.65 m\({}^{3}\)/d to 106.37 m\({}^{3}\)/d) when the well number increases from one to four.  
(3) A new evaluation model is proposed to quantitatively evaluate the permeability magnification of the fracture and reservoir when adopting the large-sized proppant. The present model shows that the fracture permeability can be increased 27-240 times in the porosity range of 0.5 \(\leq\varphi\leq\) 0.9 when using the discontinuous proppant. The present finding further confirms that there is a huge potential for applying the discontinuous large-sized proppant to enhance the reservoir permeability.  
Conceptualization, W.G., Q.C. and H.L.; methodology, W.G. and Q.C.; software, D.X. and S.Y.; validation, H.L. and D.X.; formal analysis, H.L.; investigation, D.X.; resources, W.G.; data curation, S.Y.; writing--original draft preparation, H.L.; writing--review and editing, W.G.; visualization, D.X.; supervision, W.G.; project administration, W.G.; funding acquisition, W.G. and Q.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the National Natural Science Foundation of China, grant number 51704285 and 52274103.  
Not applicable.  
The data presented in this study are available in article.  
The authors declare no conflict of interest.