Geofluids

Volume 2017, Article ID 4129240, 8 pages

https://doi.org/10.1155/2017/4129240

## Determining the REV for Fracture Rock Mass Based on Seepage Theory

^{1}Department of Earthquake Science, Institute of Disaster Prevention, Sanhe 065201, China^{2}School of Water Resources and Environment, China University of Geosciences, Beijing 100083, China

Correspondence should be addressed to Lili Zhang; moc.621@861ililgnahz

Received 9 January 2017; Revised 8 March 2017; Accepted 20 April 2017; Published 14 May 2017

Academic Editor: Shuyu Sun

Copyright © 2017 Lili Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Seepage problems of the fractured rock mass have always been a heated topic within hydrogeology and engineering geology. The equivalent porous medium model method is the main method in the study of the seepage of the fractured rock mass and its engineering application. The key to the method is to determine a representative elementary volume (REV). The FractureToKarst software, that is, discrete element software, is a main analysis tool in this paper and developed by a number of authors. According to the standard of rock classification established by ISRM, this paper aims to discuss the existence and the size of REV of fractured rock masses with medium tractility and provide a general method to determine the existence of REV. It can be gleaned from the study that the existence condition of fractured rock mass with medium tractility features average fracture spacing smaller than 0.6 m. If average fracture spacing is larger than 0.6 m, there is no existence of REV. The rationality of the model is verified by a case study. The present research provides a method for the simulation of seepage field in fissured rocks.

#### 1. Introduction

Seepage problems of the fractured rock mass have always been a heated topic within hydrogeology and engineering geology. Many problems that are closely related to the research of fracture seepages, including the stability of dam foundation and seepage, the stability of bedrock side slopes under the influence of groundwater, fissure deposit, the prevention of mine water inrush, and the leakage and diffusion of nuclear waste. However, the study of the seepage problems of the fractured rock mass is still in its preliminary stage at present. The general method regards fractured media as porous media and uses the permeability tensor of porous media to describe the seepage characteristics of the fractured media. There are different types of structural plane in fractured rock mass, which lead to the complexity of rock mass characteristics. Therefore, the study on rock mass model is always one of the important problems in rock mechanics. The representative elementary volume (REV) is the basis to determine a rock mass mechanics model, and it is necessary to research the REV of fractured rock mass effectively, so that the REV size of the fractured rock mass can be determined.

The concept of the REV was the first introduced in continuum mechanics by Bear [1], and it is to be used to describe the flow in the porous media. The parameter of interest is both homogeneous and statistically stationary, which will ensure consistency in flow simulation studies. The REV is defined in two situations on unit cell in a periodic microstructure and volume containing a very large set of microscale elements, possessing statistically homogeneous properties. The REV has been discussed by many authors [2–17]. The REV of a fractured rock mass is the smallest volume in during the study of parameter when the hydraulic conductivity is a constant value. One special concern is the evaluation of the REV of the fractured rock masses, due to the fact that fluid flow in fractured rock masses is of high scale effect [2, 18–23]. Previous studies assumed that the anisotropy was achieved by making use of different correlation lengths in the horizontal and vertical directions, and flow barriers were modeled stochastically [24–36].

Snow [37] concluded the math expressions of single and infinite fracture permeability tensors, assuming that the fracture seepages did not interfere with each other, while the overall permeability tensor of fracture network was the linear superposition of all fractures. As the fracture network was the same as the porous media, the permeability can be expressed by a permeability tensor. A fractured network can be approximately viewed as a porous medium, which was if the equivalent porous media of fractures are existent, then the permeability coefficients can be expressed by a symmetric second-order tensor. Li and Zhang [38] also conducted research on the REV of fractures. Bear [1] proved that if or in the polar coordinate system, mapping can form an ellipse (under the condition of three-dimensional ellipsoid), where represents radius vector, represents permeability coefficient in the direction of the flow line, and represents permeability coefficient in the direction of the hydraulic gradient. Long and Witherspoon [39] proposed that if the representative elementary volume of the fractured rock mass (REV) existed, the following two conditions must be met:(1)The media must be uniform in the area of study; that is, the average permeability coefficient of the area of study changes along with the scope of the study with big change; then it can be determined that this area of study is uniform.(2)In the polar coordinate system, the equivalent permeability coefficient* k* in each direction within the area of study can be described using an ellipse approximately at this time .

#### 2. Introduction to FractureToKarst Software

This paper uses discrete fracture network (DFN) software, according to the standard of rock classification established by ISRM, and discusses the existence of REV of fractured rock masses with medium tractility. The discrete fracture network (DFN) software FractureToKarst for seepage in fracture rock mass is a kind of software using the Monte Carlo method. It can generate a two-dimensional fracture network of arbitrary shapes, set the common statistical parameters of fracture, filter the fractures within the area of study, and proceed with automatic discrete can be set. The head value and equivalent permeability coefficient of each node can be calculated by using the water balance principle. In the study, set an aspect ratio for the area of study 2 : 1 [40], calculate an equivalent permeability coefficient every 10° rotation of the area of study, every area of study can get 36 equivalent permeability coefficients, and discuss whether the direction of the equivalent permeability coefficient in polar coordinates can be described as a permeability coefficient ellipse or not, thus determining the existence of REV.

##### 2.1. Mathematical Model

Mathematical model is the water dynamic model of the system. Fracture network is composed of a single, flat, smooth fracture, in a state of the laminar flow in a single fracture and can be described by the cubic law:

*Type*. is the boundary flux; is the density of water; is the gravitational acceleration; is flow dynamic viscosity coefficient; is fracture width; is two-head difference of the fracture; is the length of the fracture section.

In the fracture network, each node can establish a hydraulic link equation by the water balance principle:

*Type*. is the node flux; is the boundary flux. The hydrodynamic equations can be constructed by combining formula (1) and formula (2), and the head distribution of the fracture node within the system can be derived by the equations.

##### 2.2. Model Algorithm

Model algorithm is a numerical method. The detailed steps are as follows.

(1) Grid discretization: all the fractures can be divided into the smallest fracture section by the nodes. Each node and connected fractures make a unit balance zone, as shown in Figure 1.