## 3 Methods for Calculating the Proppant Crushing Rate and Fracture Conductivity
### Fracture Conductivity  
Fracture conductivity is the product of fracture permeability and fracture width. Considering the dimensional relationship, the fracture conductivity can be expressed as:  
\[C_{f}=0.1K_{f}w_{f}. \tag{21}\]  
where \(C_{f}\) is the fracture conductivity, \(\upmu\)m\({}^{2}\cdot\)cm; \(K\) is the fracture permeability, \(\upmu\)m\({}^{2}\); and \(w_{f}\) is the fracture width, mm. According to the Kozeny-Garman equation [7; 13; 32], the fracture permeability becomes:  
\[K=\frac{\overline{R}_{p}^{2}}{180}\frac{\phi^{3}}{(1-\phi)^{2}}. \tag{22}\]Among them:  
\[\overline{R}_{p}=\frac{\sum\limits_{i=1}^{n_{p}}R_{i}}{n_{p}}. \tag{23}\]  
where \(K\) is the fracture permeability, \(\mu\)m\({}^{2}\); \(\phi\) is the porosity, dimensionless; \(R_{i}\) is the proppant particle size (diameter), \(R_{i}\in[R_{\text{min}},\,R_{\text{max}}]\), mm; \(\overline{R}_{p}\) is the average size of the proppant, mm; \(n_{p}\) is the total number of proppant particle, dimensionless; and \(i\) is the number of proppant particles, i = 1, 2,..., \(n_{p}\). The porosity is related to the total area of the fracture and the area occupied by the proppant particle. In the next section, we discuss the two situations of the non-breakage and breakage of proppant particles separately:  
1. Situations of non-breakage (\(\eta=0\))  
When the fracture closure pressure is not applied, the proppant particles in the fracture do not break. For this situation, the porosity is the ratio of the pore area in the fracture to the total area of the fracture. Also being based on the characteristics of the proppant particles, the porosity is calculated as:  
\[\phi=\frac{w_{f}L_{f}-\sum\limits_{i=1}^{n_{p}}\left(\frac{1}{4}\pi R_{i}^{2} \right)}{w_{f}L_{f}}. \tag{24}\]  
where \(\phi\) is the porosity; \(w_{f}\) is the fracture width, mm; \(L_{f}\) is the fracture length, mm; \(R_{i}\) is the proppant particle size (diameter), \(R_{i}\in[R_{\text{min}},\,R_{\text{max}}]\), mm; \(n_{p}\) is the total number of proppant particles; and \(i\) is the number of proppant particle, i = 1, 2,..., \(n_{p}\). 2. Situations of breakage (\(\eta\
eq 0\))  
When the closure pressure is applied, the proppant particles in the fracture will break. The broken proppant particles fill the pores between the unbroken proppant particles, and the original area occupied by them causes the fracture to shrink and the width of the fracture to become smaller under the fracture closure pressure. Based on the principle of the conservation of mass, the area occupied by the proppant particles remains unchanged regardless of changes in the microscopic shape and size of the broken proppant particles. Therefore, the formula for the porosity is changed from (24) to:  
\[\phi^{\prime}=\frac{\left(w_{f}L_{f}-A_{c}\right)-\sum\limits_{i=1}^{n_{p}} \left(\frac{1}{4}\pi R_{i}^{2}\right)}{w_{f}L_{f}-A_{c}}, \tag{25}\]  
where \(A_{c}\) is the area of the broken proppant particles, that is:  
\[A_{c}=\sum\limits_{i=1}^{n_{p}}\left(\frac{1}{4}\pi R_{i}^{2}\eta_{i}\right), \tag{26}\]  
Substituting Equation (26) into Equation (25), the porosity is obtained as:  
\[\phi^{\prime}=1-\frac{\sum\limits_{i=1}^{n_{p}}\left(\pi R_{i}^{2}\right)}{4w _{f}L_{f}-\sum\limits_{i=1}^{n_{p}}\left(\pi R_{i}^{2}\eta_{i}\right)}. \tag{27}\]At the same time, the fracture width becomes smaller under the fracture closure pressure. The formula for calculating the fracture width is:  
\[w^{\prime}_{f}=w_{f}-\frac{\sum\limits_{i=1}^{n_{p}}\left(\pi R_{i}^{2}\eta_{i} \right)}{4L_{f}}. \tag{28}\]  
where \(w^{\prime}_{f}\) is the fracture width of the proppant after breakage, mm; \(w_{f}\) is the fracture width of the proppant before breakage, mm; \(L_{f}\) is the fracture length, mm; \(R_{i}\) is the proppant particle size (diameter), \(R_{i}\in[R_{\text{min}},\,R_{\text{max}}]\), mm; \(n_{p}\) is the total number of proppant particles; \(\eta_{i}\) is the crushing coefficient of the proppant; and \(i\) is the number of proppant particles, i = 1, 2,..., \(n_{p}\).