Geofluids

Volume 2018 (2018), Article ID 7535927, 22 pages

https://doi.org/10.1155/2018/7535927

## Study on the REV Size of Fractured Rock in the Non-Darcy Flow Based on the Dual-Porosity Model

^{1}Key Laboratory of Ministry of Education for Geomechanics and Embankment Engineering, Hohai University, Nanjing 210098, China^{2}College of Civil and Transportation Engineering, Hohai University, Nanjing 210098, China

Correspondence should be addressed to Yuan Wang; moc.361@uhhnauygnaw

Received 21 September 2017; Revised 10 February 2018; Accepted 19 February 2018; Published 22 April 2018

Academic Editor: Cornelius Langenbruch

Copyright © 2018 Yuan Wang 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

For the problem of whether the representative elementary volume (REV) obtained in the Darcy flow is also applicable to the case of the non-Darcy flow, the study on the REV size within the non-Darcy flow is proposed tentatively. The concept of the REV in the non-Darcy flow is based on the definition of the REV. According to the determination of the REV in the Darcy flow, the intrinsic permeability and non-Darcy coefficient are used simultaneously for the determination of the REV in the non-Darcy flow. The pore pressure cohesive element (PPCE) is developed with the subroutine in ABAQUS. Then the simulation method of the Darcy and non-Darcy flow in the fractured rock mass is built using the PPCE. The proposed plan is examined through the comparison with existing research results. It is validated that this technic is efficient and accurate in simulating the Darcy and non-Darcy flow in the fractured rock mass. Combined with fracture networks generated by Monte Carlo Simulation technique, the PPCE is applied to the study on the REV size. Both conditions of the Darcy and non-Darcy flow are simulated for comparison. The simulation results of this model show that the REV of the non-Darcy flow is inconsistent with the REV of the Darcy flow, and the REV of the non-Darcy flow is more significant than the REV of the Darcy flow. The intrinsic permeability tensors obtained in the Darcy flow and the non-Darcy flow are basically the same.

#### 1. Introduction

Hydraulic property of the fractured rock mass plays a vital role in on-site engineering, like the underground basement, oil storage, CO_{2} storage, and giant dams. The numerical simulation method can provide essential references for such projects, among which the Finite Element Method (FEM) is the most sophisticated numerical method. In the simulation of flow in the porous media, the equal intrinsic permeability should be determined first. However, as for the fractured porous media, before determining the equivalent intrinsic permeability, there is another critical parameter, the representative elementary volume (REV) to be determined. The REV is defined as the minimum volume of the sampling domain, beyond which the intrinsic permeability of the sampling domain remains essentially constant [1]. Therefore, the determination of the REV size is extremely important and meaningful for understanding the hydraulic behavior of the fractured rock mass.

In 1982, Long et al. [2] first studied the DFN (Discrete Fracture Network), in which the scale can be regarded as porous media, and two proposed qualifications: (i) the REV exists; (ii) the equivalent hydraulic property at the certain REV can be approximated by an intrinsic permeability tensor. Oda [3, 4] researched the fractured rock’s REV, anisotropy, and inherent permeability tensor using numerical simulation. Min et al. [5] investigated the REV by the Monte Carlo Simulation technique method for the generation of the DFN based on the results of a site characterization of the field and proposed the “CV” values (coefficient of variation), which is defined as the ratio of standard deviation over the mean value, and the prediction error can be adopted for the determination of the REV. Baghbanan and Jing [6] investigated the intrinsic permeability of fractured rock mass with the constant aperture when considering the correlation between the disturbed fracture aperture and the trace length. The influence of different aperture distributions on the REV is studied by comparing. Min et al. [7] and Baghbanan and Jing [8] examined the effects of stress on the intrinsic permeability and fluid flow in fractured rock mass. The stress ratio was mainly considered for the study, and the results showed that it became more difficult to establish an equivalent tensor and the REV with increasing stress ratios. It is also found that there is a significant difference between correlated and noncorrelated apertures and the fracture length distribution [8]. The studies on the REV in three dimensions have also been reported [9, 10]. The volumetric joint count calculation is applied to the determination of the REV in three dimensions [9]. Thus, in general, there have been many studies on the REV and the intrinsic permeability tensor of the fractured rock mass. However, almost all the reviews of the REV above rely on two fundamental assumptions: (i) the rock matrix is impermeable, and the fluid flow can only occur through fractures. Therefore, the DFN analysis is widely applied; (ii) the fluid flow through the fractures obeys the cubic law. It is also important to take the intrinsic permeability of the rock matrix into consideration [11]. Besides, naturally, the Darcy flow can occur at low flow rates and the fluid flow would enter the non-Darcy flow zone with the high-pressure gradient unaffectedly [10, 12–15]. Thus, considering the intrinsic permeability of the rock matrix and the non-Darcy flow in the fracture is necessary for the further study on the REV of the fractured rock mass.

In 1960, Barenblatt et al. [16] first proposed the dual-porosity model for the simulation of the fluid flow in the fractured rock mass. In the dual-porosity model, the fluid can both flow in the rock matrix and fracture. In addition, there is a fluid exchange between them, which is more realistic. Many achievements have been made [11, 16–19]. Ren et al. [20] extended a unified pipe network method (UPNM), which is a dual-porosity model, and applied it to on-site engineering. The results show that the UPNM has excellent advantages in simulating free surface flows in fractured rock slopes and oil underground storage projects with the water curtain system. Excellent reflection of the pore pressure field has been found by adopting the dual-porosity model [11, 21]. Chen et al. [22, 23] proposed the composite element method (CEM) based on the dual-porosity model. The CEM is adopted to investigate the REV of the fractured rock mass and numerical simulation of the on-site project. However, at present, the study on the dual-porosity model is mostly based on self-development, and it is quite complicated to realize this point, which makes it difficult to become popular. Therefore, it becomes more necessary for the realization of the dual-porosity model in mature commercial software, and it can exert advantages by simplifying its application in on-site engineering and related researches.

Moreover, many studies on the non-Darcy flow in the fracture have been carried out. Currently, it is considered that inertial force mainly causes the non-Darcy flow. Many experiments include it [10, 24, 25], and it is found that the fluid would present the “weak inertia regime” before totally entering the fully developed turbulence from the Darcy flow. Moreover, it is mentioned that the loss of initial inertial effect mainly results from fracture geometries. As for such inertial losses, the Forchheimer’s equation has been proved by many experiments, which describe such flow behavior very well. Besides, as for the study of flow behavior in the single fracture or intersecting fractures, many works have been done on the factors like equivalent hydraulic apertures, roughness, and pressure gradients [26]. Only a few works focused on the multifractures, especially the non-Darcy flow behavior in the fracture network and its REV size. Zimmerman et al. [27] measured and computed the intrinsic permeability of a natural sandstone with fractures. When the Reynolds number is larger than 20, both results of the experiment and numerical simulation exhibit a Forchheimer-type regime, in which the non-Darcy flow pressure drop is quadratic in the flow rate. Many non-Darcy flow experiments in the fracture network have also been conducted [28, 29]. It has been reported that when the hydraulic gradient is large enough, the inertial effect becomes nonnegligible and the nonlinear characteristic of the fluid flow should be taken into consideration necessarily [28]. What is more, the study on the Darcy and non-Darcy flow based on the dual-porosity model is rare. The 2D models, based on the dual-porosity model and mathematic analysis, are established by taking the fracture as interfaces within the Forchheimer flow while the flow in the surrounding matrix is such that Darcy’s law is adequate [30–32], but those studies only set few fractures in porous media. Generally, to take both the dual-porosity model and the non-Darcy flow behavior in the fracture network into consideration, the REV size should be further studied.

Throughout the previous literature, on one hand, most studies on the REV of the 2D fractured rock rely on the discrete model, and it is assumed that the rock matrix is impermeable and the fluid flow only occurs in the fracture. Considering that the pressure in the matrix would affect the velocity of flow in the breach, the REV size of fractured rock based on the dual-porosity model is investigated. Besides, the simulation method of the Darcy and non-Darcy flow in the fractured rock mass is built, which is conducive to promotion. On the other hand, almost all the studies on the REV involve the fluid flow in the fracture obeying the Cubic law, and as for the nonlinear flow, most studies focus on the single fracture and the fracture network. The survey on the REV in the non-Darcy flow has not been conducted in previous work. Based on the review of the non-linear flow behavior, the REV size of fractured rock in the non-Darcy flow is tentatively investigated.

In this paper, firstly, based on the dual-porosity model and the mathematics model [30], a simulation method is proposed with the pore pressure cohesive element (PPCE) in ABAQUS, which can simulate both the Darcy flow in the matrix and the non-Darcy flow in the rock fracture. According to the interpretation of the REV in the Darcy flow, the definition of the REV in the non-Darcy flow is given, when the concept of the REV in the non-Darcy flow is proposed based on the description of the REV. Then, focusing on the problem whether the representative elementary volume (REV) obtained in the Darcy flow is also suitable for the case of the non-Darcy flow, the REV size in the non-Darcy flow with the stochastic fracture network generated by the Monte Carlo method is investigated. In this paper, the REV of fracture rock in the Darcy flow is compared with that in the non-Darcy flow, when paying attention to the difference of REV sizes between the Darcy and non-Darcy flow.

#### 2. The Simulation Theory and Method of the Darcy and Non-Darcy Flow Based on the Dual-Porosity Model

##### 2.1. Darcy’s Law and Forchheimer’s Law

Considering that the velocity of the fluid in the porous rock matrix is much lower than in the fracture, the fluid flow in the matrix always obeys Darcy’s law. For the single-phase flow and incompressible flow in porous media, Darcy’s law can be expressed as (in the absence of gravity)where is the flow velocity; is the intrinsic permeability of porous media; is the pressure gradient in the porous media.

When the flow velocity is low in the fracture, the relationship between the volumetric flow rate and the pressure gradient is linear, and it can be described as the Cubic law:where is the volumetric flow rate or discharge; is the fracture aperture; is the aperture of the idealized parallel fracture, and it is the equivalent hydraulic aperture for the rough fracture; is the dynamic viscosity of the fluid; is the intrinsic permeability defined as ; is the cross-section area. When the slope of the flow rate and the pressure gradient no longer coincide, the flow would enter into the non-Darcy regime, and its constitutive relationship can be described as Forchheimer’s law:where is the pressure gradient along the flow direction; and are the coefficients, which describe the energy losses of the flow caused by viscosity and inertia, respectively. and in (3) are commonly written as follows:where is called the non-Darcy coefficient, whose value is related to the aperture and roughness of the fracture. Reynolds number Re is defined as the ratio of inertial force to viscous force, while Re can be calculated as follows:where is the density of fluid.

##### 2.2. The Pore Pressure Cohesive Element

The pore pressure cohesive element (PPCE, 6-node element) is developed from a cohesive element (4-node element), which can simulate the fluid flow in fractured rock [33]. Chen et al. [34] and Carrier and Granet [35] study hydraulic fractures with the PPCE, indicating the PPCE can simulate the fluid flow in fractured rock precisely. As shown in Figure 1, in the two-dimensional case, nodes 3, 4, 5, and 6 have not only a freedom degree of 8 (the default pore pressure of ABAQUS is 8), but also the freedom degree of 1 and 2 (the freedom degree of the displacement), whereas nodes 1 and 2 only have a freedom degree of 8. In the simulation, nodes 1, 3, and 5 (2, 4, and 6) have different pore pressure values, but six nodes have the same coordinates when the displacement is excluded, so that the entire unit looks like a line unit. The fluid flow in the PPCE includes the tangential flow within the fracture and the normal flow across the fracture, as shown in Figure 1, while Figure 2 shows the meshing examples for 2D intersecting fractures with the PPCE in ABAQUS.