# Determination of Geometrical REVs Based on Volumetric Fracture Intensity and Statistical Tests
## 6 Conclusions  
This paper presents a method for estimating the geometrical representative element volume of the fractured rock mass based on the volumetric fracture intensity P\({}_{32}\) and two statistical tests. It was demonstrated that this method was used to calculate the volumetric fracture intensity P\({}_{32}\) and determine the geometrical REV size at the Maji dam site in China.  
The field fracture data were collected by using the rectangular window sampling method. A 150 m \(\times\) 80 m \(\times\) 50 m 3D fracture network model was generated to calculate the geometrical REV size of the studied rock masses. The volumetric fracture intensity P\({}_{32}\) values were determined based on the 3D fracture network model in random regions for cube side lengths ranging from 1 to 20 m. The results show that the distribution of the P\({}_{32}\) values for each cube size is different. The size effect and spatial effect are very significant; the P\({}_{32}\) values from the same sample sizes at different locations vary; as the sample size increases, the change in P\({}_{32}\) values reduces gradually, and the fluctuation in P\({}_{32}\) values decreases significantly.  
To consider the anisotropic characteristics of a rock mass and comprehensive differences between different methods, the likelihood ratio test and Wald-Wolfowitz runs test were used to discuss the geometrical REV size of fractured rock masses. A 5% significance level was selected to estimate the statistical similarity between two sample volumes. Subsequently, the geometrical REV size was defined as 13 m using the Wald-Wolfowitz runs test method, but the geometrical REV size was defined as 12 m using the likelihood ratio test. Considering that different test methods emphasize different considerations and impact factors, the larger cube size, 13 m, was chosen. Finally, the geometrical REV size of the fractured rock mass was determined to be 13 m at the Maji dam site in China. In this study, the larger fractures were not involved in the computation, and the location of the fracture centroid is assumed to obey a homogeneous Poisson process. If the larger fractures cannot be ignored or the hypothesis is not set up, the REV size may strongly increase or cannot be found. Therefore, we should identify and deal with such cases in the future research.  
Y.L.: Article writing, 3D fracture network model establishment; Q.W.: 3D fracture network model establishment and design of the work; J.C.: 3D fracture network model establishment and design of the work; S.S.: Data collection and analysis; J.Z.: Data analysis and revising the manuscript; X.H.: Data collection and revising the manuscript.  
**Acknowledgments:** This work was supported by the key project of National Natural Science Foundation of China (Grant No. 41330636), the National Natural Science Foundation of China (Grant No. 41702301), China Postdoctoral Science Foundation (Grant No. 2017M611324) and the Graduate Innovation Fund of Jilin University (Grant No. 2017137).  
**Conflicts of Interest:** The authors declare no conflict of interest.