# Studies of Fracture Damage Caused by the Proppant Embedment Phenomenon in Shale Rock
## Materials and Methods
### Methodology for Studying the Embedment Phenomenon
The percentage damage of the fracture surface (\(PDW_{e}\)) for the profile, along the measurement section was determined according to Equation (5) [7; 10; 20; 21], expressed in (%):  
\[PDW_{e}=\frac{\sum_{i=0}^{n}W_{e_{i}}}{L}\cdot 100 \tag{5}\]  
where \(We_{i}\) is the valley width, i.e., the embedment of a proppant grain in the fracture face (mm), and \(L\) is the length of the measurement section (mm). The total percentage damage of the fracture surface \(PDW_{e}\) (embedment of the embedding proppant grains on the surface of the fracture faces) was determined according to Equation (6) [7; 10; 20; 21], expressed in (%):  
\[PDW_{e_{i}}=\frac{PDW_{e_{T,a}}+PDW_{e_{B,a}}}{2} \tag{6}\]  
where \(PDW_{e_{T,a}}\) is the average percentage damage of the surface of the top fracture face (rock), corresponding to the arithmetic average of the obtained values for individually specified areas (%), and \(PDW_{e_{B,a}}\) is the average percentage damage of the surface of the bottom fracture face (rock), corresponding to the arithmetic average of the obtained values for individually specified areas (%). The effect of the embedment phenomenon on the effective width of the fracture packed with proppant after exposure to axial compression stress was determined using Equations (7) and (8) [7; 10; 20; 21; 28]:  
Figure 4:0  
where \(W_{f}\) is the fracture width packed with proppant, taking into account the embedment phenomenon (mm), and \(W_{fm}\) is the maximum fracture width packed with proppant, without the occurrence of the embedment phenomenon (mm). The percentage reduction of the fracture width (\(PRW_{f}\)) packed with proppant, taking into account the embedment phenomenon, was determined according to Equation (8) [7; 10; 20; 21], expressed in (%):  
Figure 4:1  
The maximum width \(W_{fm}\) of the fracture packed with proppant, without the occurrence of the embedment phenomenon, was determined according to the research procedure previously mentioned in this paper. The only difference was the use of cylindrical steel plugs instead of cylindrical core plugs, which have a steel hardness of more than 43 on the Rockwell C scale (HRC). The maximum width \(W_{fm}\) of the fracture packed with proppant was measured throughout the testing with the use of an LVDT (Linear Variable Differential Transformer) device. LVDT readings took into account the amount of deformation of the test unit (i.e., hydraulic press, measuring chamber, and steel plugs) under the specified conditions of axial compressive stress and temperature. Execution of a Laboratory Simulation of the Embedment Phenomenon and Analysis of the Obtained Test Results  
The tests were performed on cylindrical core plugs with a diameter of 2.54 cm. Firstly, the average primary roughness \(Ra\) of the entire surface of the core plug face (for the top and bottom fracture face), presented in Figure 7, was determined according to the test procedure described in the previous part of the paper. It was determined as an arithmetic average of two selected areas on the face of the tested core plug, from one profile running across the tested area. These tests were performed using an optical microscope (Figure 8), and the results are presented in Figures 9 and 10. Figure 4:2 Optical microscope with 3D software. Figure 4:3 Determination of the primary roughness of the core plug face (fracture faces): (a) Top; (b) Bottom. Figure 4:4 Surface of the core plug sample with a diameter of 2.54 cm before the embedment test: (a) Top; (b) Bottom. Figure 4:5 Average depth \(H_{e_{T,a}}\) and average percentage surface damage \(PDW_{e_{T,a}}\) for the top core plug (Test 1): (**a**) Area 1; (**b**) Area 2. Figure 4:6 Average depth \(H_{e_{k}a}\) and average percentage surface damage \(PDW_{e_{k}a}\) for the bottom core plug (Test 1): (**a**) Area 1; (**b**) Area 2. Test no. 2 was performed in order to determine the maximum achievable width of the fracture packed with a light ceramic proppant without embedment.