Geofluids

Volume 2018, Article ID 9352608, 14 pages

https://doi.org/10.1155/2018/9352608

## Effects of Hydraulic Gradient, Intersecting Angle, Aperture, and Fracture Length on the Nonlinearity of Fluid Flow in Smooth Intersecting Fractures: An Experimental Investigation

^{1}The Key Laboratory of Safety for Geotechnical and Structural Engineering of Hubei Province, School of Civil Engineering, Wuhan University, Wuhan 430072, China^{2}College of Architecture and Civil Engineering, Beijing University of Technology, Beijing 100124, China^{3}Department of Geological Engineering, School of Geosciences and Info-Physics, Central South University, Changsha, Hunan 410083, China

Correspondence should be addressed to Lifeng Fan; nc.ude.tujb@gnefilnaf

Received 12 September 2017; Revised 23 November 2017; Accepted 11 January 2018; Published 7 March 2018

Academic Editor: Yuan Wang

Copyright © 2018 Zhijun Wu 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

This study experimentally investigated the nonlinearity of fluid flow in smooth intersecting fractures with a high Reynolds number and high hydraulic gradient. A series of fluid flow tests were conducted on one-inlet-two-outlet fracture patterns with a single intersection. During the experimental tests, the syringe pressure gradient was controlled and varied within the range of 0.20–1.80 MPa/m. Since the syringe pump used in the tests provided a stable flow rate for each hydraulic gradient, the effects of hydraulic gradient, intersecting angle, aperture, and fracture length on the nonlinearities of fluid flow have been analysed for both effluent fractures. The results showed that as the hydraulic gradient or aperture increases, the nonlinearities of fluid flow in both the effluent fractures and the influent fracture increase. However, the nonlinearity of fluid flow in one effluent fracture decreased with increasing intersecting angle or increasing fracture length, as the nonlinearity of fluid flow in the other effluent fracture simultaneously increased. In addition, the nonlinearities of fluid flow in each of the effluent fractures exceed that of the influent fracture.

#### 1. Introduction

The nonlinear behaviour of fluid flow in natural fractured rock masses is a critical hotspot in numerous branches of geoscience and rock engineering fields, such as geological disposal of radioactive waste [1, 2], reservoir storage [3, 4], underground mining [5, 6], and geothermal extraction [7, 8]. Considering that an intact rock matrix in a deep formation has a very low permeability (which is usually assumed to be impermeable) [9–11], the nonlinear behaviour of fluid flow in fractured rock masses is heavily controlled by the flow behaviour in a single fracture or fracture network. In recent decades, the effects of fracture and hydraulic characteristics, such as fracture length, fracture density, aperture (or hydraulic aperture), fracture orientation, fracture connectivity ratio, hydraulic gradient, intersecting angle, surface roughness, scale effect, and number of intersections and dead-ends, on the nonlinear behaviour of fluid flow in rock fractures have been extensively and systematically studied, forming the basis for understanding nonlinear fluid flow behaviour in natural fractured rock masses. However, due to the complexity of fracture distribution and fracture characteristics in naturally fractured rock masses, clearly determining the effects of all the fracture and hydraulic parameters on the nonlinear behaviour of fluid flow in naturally fractured rock masses is still challenging.

Many researchers have focused on the effects of the fracture and hydraulic characteristics on the fluid flow behaviour in a single fracture, intersecting fractures, and fracture networks. Long and Billaux [10] presented a fracture network model that accounted for the observed spatial variability by generating a network in subregions, where the properties of each subregion were predicted through geostatistics. They found that approximately 0.1% of the fractures primarily controlled the permeability of the system. De Dreuzy et al. [11, 12] conducted a numerical and theoretical study on the permeability variation by assuming that the fracture length obeys a power law distribution and concluded that the hydraulic properties of fracture networks with a power law length distribution can be classified into three simplified types. When the power law exponent , fracture networks essentially consist of very small fractures and the percolation theory applies. On the other hand, when , flow is mostly channelled into longer fractures and fracture networks can be considered as a superposition of long fractures. Between these ranges (), fracture length distributions cannot be restricted to a unique length, even though the smaller fractures have a small contribution to flow. Olson [13] focused on a nonlinear aperture-to-length relationship and suggested that fracture apertures scale with their lengths to the 1/2 power; this result was obtained by using linear elastic fracture mechanics in a homogeneous body, with subcritical and critical (equilibrium law) fracture propagation criteria. In addition, Olson [13] determined that a fracture aspect ratio (aperture/length) decreases with increasing fracture length to the negative 1/2 power. Min and Jing [14] conducted a series of numerical simulations of the mechanical deformation of fractured rock masses at different scales with many realizations of Discrete Fracture Networks (DFNs) and concluded that a representative elementary volume can be defined and the elastic properties of the fractured rock mass can be approximately represented by the elastic compliance tensor. Min et al. [15] investigated the stress-dependent permeability issue in fractured rock masses, considering the effects of nonlinear normal deformation and shear dilation of fractures. They found that the permeability of fractured rocks decreases with increasing stress magnitudes when the stress ratio is not sufficiently large to cause shear dilation of the fractures, whereas the permeability increases with increasing stress when the stress ratio is sufficiently large. In addition, permeability changes at low stress levels are more substantial than those at high stress levels due to the nonlinear relation between fracture normal stress and displacement. Based on a newly developed correlation equation, Baghbanan and Jing [16] investigated the permeability of fractured rocks by considering the correlation between the distributions of aperture and trace length of the fractures. Their results showed that there is a significant difference between correlated and noncorrelated apertures and fracture length distributions, which demonstrated that the hydromechanical behaviour of fractured rocks is considerably scale- and stress-dependent when the aperture and fracture length distributions are correlated. By solving the Navier-Stokes equations without linearization, using a self-developed 2D finite volume method, Zou et al. [17] investigated the effects of wall surface roughness on fluid flow through rock fractures. Their results indicated that even with the same total flow rate, the flow patterns and velocity fields have significant differences, which are caused by the secondary roughness and can even create time-dependent dynamic flow, with moving and changing eddies, when the Reynolds number (Re) is high. Furthermore, Zou et al. [18, 19] studied the fluid flow and solute transport in a 3D rock fracture-matrix system, which had two rough-walled fractures with an orthogonal intersection. Zhou et al. [20] experimentally investigated the nonlinear flow characteristics of fluid flow through the rough-walled fractures subjected to a wide range of confining pressures (1.0–30.0 MPa/m) at low Re. They found that the obtained critical Reynolds number versus confining pressure curves generally displays a nonlinear weakening stage (I) in the early stage of confining pressure loading, which is followed by a nonlinear enhancement stage (II) as the confining pressure further increases. Using numerical simulations based on DFNs, Liu et al. [21, 22] estimated the effects of fracture intersections and dead-ends on nonlinear flow and particle transport in 2D DFNs and demonstrated that wider fracture apertures, rougher fracture surfaces, and greater numbers of fracture intersections in a DFN may result in the onset of nonlinear flow at a lower critical hydraulic gradient. In addition, they found that the effects of fracture dead-ends on fluid flow are negligible (<1.5%); however, fracture dead-ends have a strong impact on the breakthrough curves of particles in DFNs with a relative time deviation rate in the range of 5–35%. Based on fluid flow tests and numerical simulations, Li et al. [23] found that the nonlinear degree of hydraulic gradient () versus flow rate () increases with and the joint roughness coefficient (JRC), whereas the nonlinear degree of hydraulic gradient versus flow rate increases as the radius of the truncating circle decreases (the radius is equivalent to fracture length). In addition, they found that the intersecting angle affects the effluent fracture flow rate ( and stand for two effluent fractures) for three different intersecting angle patterns.

Previous studies show that a numerical model based on discrete fracture networks (DFNs), which usually consists of thousands of fracture elements and nodes [14, 21, 22], was mainly used to investigate the nonlinear fluid flow in fractured rock masses. However, a reasonable description of the fluid flow in the basic elements and nodes is the key prerequisite to ensure the accuracy of the numerical results. Reasonable modelling of nonlinear fluid flow behaviour through an entire fractured rock mass depends on realistically describing the nonlinear fluid flow in both a single fracture and an intersecting fracture network. However, the fluid flow through one-inlet-two-outlet fracture patterns with a single intersection, which is a fundamental element of DFNs, has not yet been extensively studied. Though a few studies have been carried out by numerical modelling [23] to investigate the effects of parameters such as , intersecting angle (), aperture (), and fracture length (*Rr*) on the nonlinear behaviour of fluid flow, the reliability of the numerical results is low without experimental validation. In this study, a fluid flow test system was built to conduct a series of laboratory tests on one-inlet-two-outlet specimens with particular attention paid to the effects of , , , and* Rr* on the nonlinearity of fluid flow in both effluent fractures at large Re and large . During the tests, the syringe pressure gradient was controlled and varied in the range of 0.20–1.80 MPa/m, resulting in a high flow velocity at the inlet. The outlet of each effluent was open, where the discharged fluid was collected in a bucket and later measured by an electronic balance. Based on the collected fluid weight, the flux of each effluent was calculated at the end of the experiment by dividing the effluent quantity by the collecting time.

#### 2. Methodology

##### 2.1. Theoretical Background

###### 2.1.1. Linear Darcy Flow Zone

The simple parallel plate model is the only fracture model available to calculate the hydraulic conductivity of a fracture. This model assumes a steady incompressible Newtonian fluid flow in a single fracture under a one-dimensional pressure gradient between two smooth parallel plates separated by an aperture [24, 25]. At sufficiently low , this calculation yields the well-known “cubic law” [26, 27] and “Darcy’s law” [28, 29].where is the total volumetric flow rate, which represents the flow rate of the influent fracture in this study, is the pressure gradient, is the width of the fracture, is the uniform aperture of the idealized smooth fracture, is the dynamic viscosity,* k* is the intrinsic permeability defined as [5, 27], and is the cross-sectional area equal to . Equation (1) predicts that is proportional to the cube of and that is linearly correlated with when Re is sufficiently small. The cubic law has been widely applied to rough-walled fractures in rock; however, in these applications, is replaced by the so-called hydraulic aperture [24, 30, 31]. In this paper, is equal to since the walls of the fractures in the test specimen are smooth.

###### 2.1.2. Non-Darcy Flow Zone

Darcy’s law and the cubic law are only valid when the inertial force is negligible, compared with the magnitude of the viscous force, and this condition is anticipated only when is low. Nonlinear flow occurs when, with an incremental increase in Re, increases more than a proportional incremental amount [26, 32–35]. Forchheimer’s law [8, 36–38] has been most widely used to describe the nonlinear flow in fractures and porous media, especially in strong inertial regimes:where and are coefficients, and , is the fluid density, and is called either the non-Darcy coefficient or the inertial resistance [5, 39, 40].

The hydraulic gradient is proportional to :where and are the hydraulic heads at each end of a fracture, and are the pressures at each end of a fracture, and is the effective path under the hydraulic gradient. Equation (2) can be written in a Forchheimer’s law form as follows:The term represents the linear gradient, and the term represents the nonlinear gradient.

Coefficients and in (4) are commonly written as follows:Equation (5a) states that is linearly proportional to the reciprocal of the cube of . If the values of and are measurable and constant, the value of can be calculated directly. In addition, in some studies, the value of is considered the intrinsic permeability of the fractured media [29, 41, 42]. The coefficient in (5b) is determined from the geometrical properties of the fractures and media in the experiments [34, 40, 43].

For fluid flow within fractures, the ratio of the inertial force to the viscous force is defined as the Reynolds number (Re), which is expressed as follows [44, 45]: where is the bulk flow velocity in the fractures. Critical values of Re can distinguish between the linear and nonlinear flow conditions [20, 22, 24, 31, 46, 47].

The vector flow velocity can be reduced to a one-dimensional flow velocity along a fracture intersection:

##### 2.2. Experimental Setup

###### 2.2.1. Specimen Preparation

First, specimens containing one inlet and two outlets were designed by AutoCAD, which can accurately create a regular single intersection of three fractures. Based on the control variate method, the specimens were designed with *θ* = 30°, 60°, 90°, 120°, and 150°, = 2, 3, 5, 10, and 20 mm, and* Rr* = 40, 100, 200, 300, and 400 mm. The specimens were made with PMMA (polymethyl methacrylate) material, which is transparent and easy to process. Then, the specimen design diagrams, in EPS format, were transferred to the computer of the dedicated PMMA machine. Next, the specimens were created at a series of scales with a small error: ±0.02 mm. The sizes of ,* Rr*, and were precisely measured by a digital Vernier caliper. Note that the 5.7 mm width was controlled by the length of the drill employed to cut the PMMA. The error of was ±0.1 mm (i.e., = 5.648–5.745 mm) because the surface of the initial PMMA slab was not flat. Next, a high-pressure flame gun was used to polish the fracture walls, so that the initial condition of smooth fracture walls was met in this study. Figure 1 shows the pictures of the specimens. Finally, the specimens were glued to plates with the same side area as the specimens using the PMMA adhesive. This procedure was repeated to ensure absolute sealing of the fractures.