Geofluids

Volume 2017 (2017), Article ID 2176932, 17 pages

https://doi.org/10.1155/2017/2176932

## A Review of Critical Conditions for the Onset of Nonlinear Fluid Flow in Rock Fractures

^{1}State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China^{2}State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China^{3}School of Engineering, Nagasaki University, Nagasaki 8528521, Japan

Correspondence should be addressed to Richeng Liu

Received 9 April 2017; Accepted 9 May 2017; Published 6 July 2017

Academic Editor: Zhien Zhang

Copyright © 2017 Liyuan Yu 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

Selecting appropriate governing equations for fluid flow in fractured rock masses is of special importance for estimating the permeability of rock fracture networks. When the flow velocity is small, the flow is in the linear regime and obeys the cubic law, whereas when the flow velocity is large, the flow is in the nonlinear regime and should be simulated by solving the complex Navier-Stokes equations. The critical conditions such as critical Reynolds number and critical hydraulic gradient are commonly defined in the previous works to quantify the onset of nonlinear fluid flow. This study reviews the simplifications of governing equations from the Navier-Stokes equations, Stokes equation, and Reynold equation to the cubic law and reviews the evolutions of critical Reynolds number and critical hydraulic gradient for fluid flow in rock fractures and fracture networks, considering the influences of shear displacement, normal stress and/or confining pressure, fracture surface roughness, aperture, and number of intersections. This review provides a reference for the engineers and hydrogeologists especially the beginners to thoroughly understand the nonlinear flow regimes/mechanisms within complex fractured rock masses.

#### 1. Introduction

Rock fracture network controls the main paths of fluid flow and contaminant migration in deep underground, and the estimation of permeability of fractured rock masses has been extensively studied during the past several decades in many geoengineering and geosciences such as CO_{2} sequestration, enhanced oil recovery, and geothermal energy development [1–9]. The fluid flow in rock fractures and/or fracture networks is commonly assumed to obey the cubic law, in which the flow rate is linearly proportional to the pressure drop [10–16]. However, in the karst systems and/or in the vicinity of wells during pump tests, when the flow rate/velocity is large, fluid flow enters the nonlinear flow regime and flow rate is nonlinearly correlated with pressure drop [14, 17–20]. In such case, using the cubic law to calculate fluid flow will overestimate the conductivity of rock fractures and/or fracture networks [21–24]. Therefore, a thorough understanding of the nonlinear flow properties of fluid within fractures contributes to accurately predicting permeability of fractured rock masses [25].

Previous studies have reported that there are three representative types of nonlinear flow behaviors in rock fractures induced by inertial effect, fracture dilation, and solid-water interaction [20, 26, 27], and the present study focuses on the nonlinear flow behaviors induced by inertial effect. Many factors can affect the magnitude of permeability of fractured rock masses, including fracture length [28–31], aperture [32–34], surface roughness [35, 36], dead-end [37], number of intersections [38, 39], hydraulic gradient [40], boundary stress [41, 42], anisotropy [43–46], scale [47–50], stiffness [51], coupled thermo-hydro-mechanical-chemical (HTMC) processes [52–55], and precipitation-dissolution and biogeochemistry [56]. The discrete fracture network (DFN) model, which can consider most of the above parameters, has been increasingly utilized to simulate fluid flow in the complex fractured rock masses [57–60], although it cannot model the aperture heterogeneity of each fracture [61–63]. In the numerical simulations and/or analytical analysis, the linear governing equation such as the cubic law is solved to simulate fluid flow in fractures by applying constant hydraulic gradients () on the two opposing boundaries, such as [57, 64–68], [41], [69, 70], and = unknown constants [11, 34, 46, 71–73]. This assumption that fluid flow obeys the cubic law is suitable for characterizing hydraulic behaviors of deep underground engineering, in which the flow rate is sufficiently small. For the fractured rock masses such as the karst system and/or the in situ hydraulic tests, the flow rate is relatively high and nonlinearly correlated with the pressure drop [17, 74]. With increasing pressure drop, the fracture network permeability decreases. To accurately simulate the nonlinear flow in fractures, the Navier-Stokes (NS) equations should be solved, which are composed of a set of coupled nonlinear partial derivatives of varying orders [21, 75, 76] and are more complex than solving the cubic law. When fluid flow is in the linear regime, each fracture in the two-dimensional DFNs is represented using a line segment [77], whereas when fluid flow enters the nonlinear regime and the NS equations are solved, the real geometry (void space) of each fracture that is formed with two walls should be incorporated, which to some extent increases the difficulty of establishing the models [40, 78]. As a result, both the yearly published papers and yearly cited times with the keywords “nonlinear flow” and “rock mass” are much smaller than those with the keywords “linear flow” and “rock mass,” as shown in Figure 1. With the development of computing power, more researches are contributing to the nonlinear flow characteristics of fractures, which needs solving the NS equations and is often time-consuming. Therefore, it is a vital issue about how to determine the critical condition (i.e., critical Reynolds number or critical hydraulic gradient) for the onset of nonlinear flow.