Investigation of Nonlinear Flow in Discrete Fracture Networks Using an Improved Hydro-Mechanical Coupling Model

Abstract

Fractures commonly exist in rock masses; the coalescence of fractures provides fluid flow pathways in a fractured rock mass and greatly increases the flow capacity of fractured rock. This work aims to study the characteristics of nonlinear flow in fractures. A series of tests were conducted and indicated that the Forchheimer law performed well when describing the nonlinear relationship between hydraulic gradient and flow. The test results also indicate that higher water pressure may induce stronger nonlinearity. Additionally, the linear and nonlinear coefficients of the Forchheimer law increase with a decrease in the particle size of the filling material in fractures. On the basis of the laboratory results, the classical Forchheimer law was modified by considering the influence of stress on the variation of fracture aperture. A hydro-mechanical coupling model for fractured rock masses was built and programmed with a subroutine through ABAQUS. Furthermore, a random discrete fracture network was generated and simulated to prove that a high flow velocity will result in a nonlinear flow, not only in a single fracture, but also in a fracture network. The numerical results from fractured rock masses show that a ratio of the flow to the hydraulc gradient will change the flow from linear to weak nonlinearity and, finally, to strong nonlinearity with an increase in the hydraulic gradient. It also shows that the linear and nonlinear coefficients increase with an increase in the confining pressure and that they decrease with an increase in the aperture. Due to the complexity of fracture channels, a nonlinear flow is likely to occur in a fractured rock mass. Finally, the developed model was applied to simulate the flow behavior of underground engineering; the results show that the smaller the hydraulic aperture is, the higher the water pressure is required to be in order to change the flow regime from linear to nonlinear.

1. Introduction

Understanding the hydro-mechanical coupled behavior of a fractured rock mass is crucial for investigation of the rock mass properties encountered in underground engineering, such as in tunnel excavation [1,2,3], reservoir storage [4,5,6,7], and geothermal energy development [8,9,10,11,12,13,14], as well as the safety assessment of high-level radioactive waste repositories [15,16]. The interconnection of the fracture network in the rock mass provides a convenient path for fluid flow and gradually increases the flow capacity of the fractured rock mass. Therefore, fluid flow in fracture networks is important for understanding stability during the construction of rock engineering projects in fractured rock masses.

A fractured rock mass is a complex geometric system. The geometric distribution of fractures can be obtained by field surveying and mapping outcrops in a limited area, as well as logging in a well with a limited borehole and depth [17,18,19,20]. To represent the underground fracture system more realistically, the usual method is to assume the statistical distribution of the geometric parameters of the fractures and use that information in discrete fracture networks (DFNs) modelling [21,22,23]. In recent years, many scholars have studied the influence of fracture networks on the stability of underground engineering [24,25]. Notably, each generation of DFNs must strictly follow the statistical distribution law of each geometric parameter in the case study.

Generally, porous media can be divided into various types by their application scope [26,27,28,29]. It makes sense to investigate the effect of mechanical characteristics on other characteristics and performance, and the improvement of a numerical method for fractured porous media is especially significant in geoscience applications [30,31]. When the flow rate is small or the flow velocity is slow in a rock mass, a fracture can be simplified into a parallel plate model with a smooth surface, and the regularity of fluid flow obeys the cubic law. The permeability of the fractured rock mass is closely related to the aperture, roughness, and stress boundary conditions. Experimental and numerical studies carried out by Min and Kumura et al. showed that the larger the aperture, and the smaller the roughness, the greater the permeability of the fracture will be [32,33]. Extensive works related the influence of interface contact and gouge production and plastic deformation of asperities on the permeability evolution were published in the literature [34,35]. Permeability alterations due to a stress-induced closure of mated fractures were also studied. Vogler et al. indicated that the closure of a fracture reduced the fracture conductivity [36]. However, Selvadurai showed that the permeability of rock in a failure state under the action of stress was significantly improved through tests [37]. Permeability evolution during the shear process was investigated through theoretical models, laboratory experiments, and numerical simulations. These works studied the relationship between fracture geometric evolution and the change in flow behavior during shear. The results showed that the fracture expansion, in the early shearing stages, significantly increased the permeability of a fracture [38,39]. However, some studies also show that the fracture was filled with the formatted gouge, due to asperity damage while the shear displacement was large, resulting in a gradual decrease of permeability [40]. In recent decades, many studies have focused on modifying the cubic law, for example, by determining mathematical expressions that calculate the hydraulic aperture while considering factors such as the fracture surface roughness, contact ratio, and curvature [41,42,43,44]. However, in the case of high velocities, the relationship between hydraulic and flow become nonlinear; the cubic law would be no longer suitable, and the hydraulic characteristics predicted by cubic law would be quite different from the experimental results.

Previous laboratory experiments and numerical simulation results studies showed that the nonlinear flow in a fractured rock mass was mainly caused by inertial force; a larger Reynolds number resulted in a significantly nonlinear flow under a high flow velocity. The critical Reynolds number marked the beginning of the flow’s conditioning from linear to nonlinear [45], and a rougher fracture surface lead to a larger critical Reynolds number [46]. Some studies also showed that the critical Reynolds number increases with an increase in shear displacement [47]. Information about the Reynolds number under different boundary conditions was also obtained through experiment and numerical simulation [48,49]. A larger aperture, coarser fracture surface, and more fracture intersections in the DFNs will lead to nonlinear flow at a lower hydraulic gradient [50]. Wang et al. found that a nonlinear flow was enhanced significantly by the secondary roughness, which results in an earlier onset of nonlinearity [51]. An experiment carried out by Fan et al. indicated that different intersection angles of fractures will affect the distribution of an outlet flow, thus affecting the occurrence of a nonlinear flow [52]. Investigations into the linear and nonlinear coefficients of the Forchheimer law have been conducted based on a series of flow experiments and numerical simulations for rough fractures. Experimental studies showed that a higher confining pressure and a smaller fracture aperture result in an increase in the linear and nonlinear coefficients of the Forchheimer law, and the studies established the mathematical relationship between the nonlinear coefficient and hydraulic aperture [53,54]. Xiong et al. studied the nonlinear behavior of a three-dimensional fracture network by numerical simulation; the study results show that the linear and nonlinear coefficients of the Forchheimer law decrease with an increase in the percolation density of the fracture network and that they increase with an increase in the fracture surface roughness [55]. A laboratory and numerical investigation also shows that the linear and nonlinear coefficients of the Forchheimer law decrease with an increase in the shear displacement [56,57,58].

However, previous experimental studies have only focused on the influence of stress on the nonlinear flow characteristics of a single fracture, and numerical studies have hardly considered the influence of stress on the fracture aperture; therefore, the study of the influence of stress on the nonlinear flow characteristics of DFNs is still insufficient. The purposes of this study are: (1) to present the nonlinear flow of jointed rock masses through laboratory flow tests, theoretical models, and numerical simulations, (2) to develop the classical Forchheimer law for consideration of the influence of stress on the variation of fracture aperture; (3) to build a hydro-mechanical coupling model for fractured rock masses, programmed with a subroutine through ABAQUS; and (4) to evaluate the application of the proposed model in tunnel engineering to predict the water inflow during tunnel construction.

2. Laboratory Tests on Nonlinear Flow of Fractures

2.1. Sample Preparation

In this study, fractures in granite were taken to investigate nonlinear flow characteristics; the samples were cored with a diameter of 50 mm and length of 70 mm, due to the limitation of the apparatus.

To obtain a fractured granite sample, as shown in Figure 1a, firstly, an intact sample of the size suitable for the equipment was prepared; the sample was then placed on the mold, and it was ensured that the burins on the top and bottom were in contact with the sample. Then, by applying the vertical loading after fixing the sample at the central position of the mold via the screws on both sides, it was preloaded. Next, the screws were released to allow the sample to be in an unconfined state. Finally, vertical loading continued to be applied until the sample was split into two halves, as shown in Figure 1b. The planarity after splitting was different because of the different internal mineral distribution, which resulted in different roughnesses of the fracture surface. The fractured sample is shown in Figure 2, and the three samples are named G1, G2, and G3, respectively.

Through three-dimensional scanning of these fracture surfaces, a large number of point sets of three-dimensional coordinates can be obtained. Based on these point sets, a visual image of the fracture surface can be obtained (Figure 3), and the joint roughness coefficient (JRC) of the fracture surface can be calculated according to the following formulas [59,60]:

$${\mathrm{Z}}_2={\left[\frac{1}{(\mathrm{n}-1){(\Delta \mathrm{x})}^2}{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}-1}({\mathrm{Z}}_{\mathrm{i}+1}-{\mathrm{Z}}_{\mathrm{i}}}{)}^2\right]}^{1/2}$$

(1)

$$\mathrm{J}\mathrm{R}\mathrm{C}=32.69+32.98\times \lg {\mathrm{Z}}_2$$

(2)

where ${\mathrm{Z}}_2$ is the root mean square slope of the profiles based on the extracted data, Z_i represents the coordinates of the fracture surface profile, $\mathrm{n}$ is the number of data points, and $\Delta \mathrm{x}$ is the interval between the data points. Specifically, the mean JRC values of Samples G1, G2, G3 were 13.01, 13.50, and 13.97, respectively.

The test schemes are shown in

Table 1. The flow test cases.

	Sample	Diameter (mm)	Length (mm)	Filling Conditions (mesh)	Water Pressure (MPa)
Case 1	G 1	47.92	70.52	/	0.5–3.0
Case 2				50	0.1–0.5
Case 3				100
Case 4				200
Case 5	G 2	47.88	70.67	/	0.5–3.0
Case 6				50	0.1–0.5
Case 7				100
Case 8				200
Case 9	G 3	47.65	70.32	/	0.5–3.0
Case 10				50	0.1–0.5
Case 11				100
Case 12				200

. Each sample was used to conduct two tests: unfilled and filled with quartz sand for the consideration of the real state of fractures in practical engineering [61,62]. For the filled cases, the diameters of the quartz sand were 50 mesh, 100 mesh, and 200 mesh, respectively. In order to ensure that the same filling quality was used under each operating condition, 0.3 g of each filling material was weighed accurately. Then, the filling material was evenly placed on the fracture surface by hand or using a small brush. The sample was weighed pre- and post- filling to ensure that no filling material was lost during the process and that the filling amount was the same under each condition. Care was taken to ensure that the fillings were laid evenly on the fracture surface. The sample was wrapped with adhesive tape to prevent the leakage of quartz sand during the placement process of the samples, as shown in Figure 4. During the experiments, permeable stones were placed at both ends of the sample to prevent the quartz sand from being washed away.

2.2. Methods and Procedures

The apparatus was equipped with three independent loading systems of axial pressure, confining pressure, and osmotic pressure, all of which were controlled by ISCO pumps. Each rock sample was tightly wrapped in a rubber sleeve under confining pressure to ensure that the water only passed through the fractures in the sample. Figure 5 shows a schematic diagram of the apparatus.

The steady-state method was selected for measuring high permeability tests. The water pressure was pumped in from the left side of the sample and flowed out from the right side of the sample after passing through the fracture. Five water pressures (0.1 MPa, 0.2 MPa, 0.3 MPa, 0.4 MPa, and 0.5 MPa) were chosen as the input pressures for the filled fractures, and another five water pressures (0.5 MPa, 1.0 MPa, 1.5 MPa, 2.0 MPa, and 2.5 MPa) were chosen as the input pressures for the unfilled fractures. The right end of a samples maintained drained conditions. The hydrostatic pressure and the confining pressure were recorded by pressure sensors, and a balance was set at the flow outlet to record the flow at any time. All tests were carried out at a constant temperature of 25 °C. The confining and axial pressures are both maintained at 5 MPa.

2.3. Experimental Results and Analysis of the Nonlinear Flow of Fractures

When the flow velocity is small or at a low Reynolds number, the fluid flow conforms to the cubic law [63]:

$$\mathrm{Q}=-\frac{{\mathrm{w}\mathrm{e}}^3}{12\mathsf{\mu }}\nabla \mathrm{P}$$

(3)

where w is the width of the fracture, $\mathrm{e}$ is the aperture of the fracture, $\mathsf{\mu }$ is the dynamic viscosity coefficient of the fluid, $\mathrm{Q}$ is the volume flow, and ∇P is the water pressure gradient.

When the flow velocity is increasing, the inertial effects induced by the contact areas and surface roughness of the fracture should be considered, and the Forchheimer law is adopted to describe its nonlinear term [64]:

$$-\nabla \mathrm{P}=\mathrm{A}\mathrm{Q}+\mathrm{B}{\mathrm{Q}}^2$$

(4)

where $\mathrm{A}=12\mathsf{\mu }/\mathrm{w}{\mathrm{e}}^3$ and $\mathrm{B}=\mathsf{\beta }\mathsf{\rho }/{\mathrm{w}}^2{\mathrm{e}}^2$ are the linear and nonlinear coefficients, respectively; $\mathsf{\mu }$ is the dynamic viscosity coefficient of the fluid; $\mathsf{\beta }$ is the Forchheimer coefficient; $\mathsf{\rho }$ is the fluid density; $\mathrm{w}$ is the width of the fracture; and $\mathrm{e}$ is the aperture of the fracture.

The Reynolds number provides a method of comparison for the ratio of inertial force to viscous force; to quantify the relative importance of two forces under given flow conditions, the Reynolds number is defined as [65]:

$$\mathrm{Re}=\frac{\mathsf{\rho }\mathrm{v}\mathrm{e}}{\mathsf{\mu }}=\frac{\mathsf{\rho }\mathrm{Q}}{\mathsf{\mu }\mathrm{w}}$$

(5)

where $\mathsf{\rho }$ is the fluid density, v is the flow velocity, w is the width of the fracture, $\mathrm{e}$ is the aperture of the fracture, and μ is the dynamic viscosity coefficient of the fluid.

The critical Reynolds number for the transition from linear flow to nonlinear flow can be calculated by follow equation [65,66]:

$${\mathrm{Re}}_{\mathrm{C}}=\frac{\mathrm{A}\mathsf{\rho }}{9\mathrm{B}\mathsf{\mu }\mathrm{w}}$$

(6)

Flow experiments have been carried out for fracture samples with different filling conditions. All experimental results show that the relationship between the water pressure gradient $\nabla \mathrm{P}$ and volume flow Q is strongly nonlinear and the Forchheimer law could describe the results well. The correlation coefficient R² could reach more than 0.99. Figure 6 shows that an increase in flow velocity leads to a nonlinearity of flow. Non-Darcy flow is caused by non-negligible inertial damage due to a change in flow direction or velocity along the flow path, and a greater velocity results in more significant losses of inertia. A decreasing proportion in the water pressure gradient is greater than an increasing proportion of flow due to the inertial losses, and the relationship between the water pressure gradient and flow is no longer linear [1,53].

Table 2. Fitting results of coefficient of the nonlinear equation.

Filling Condition	Sample	Linear Coefficient A	Nonlinear Coefficient B	R2
With 50 mesh particles	G1	3.02 × 105	1.69 × 1012	>0.99
	G2	3.10 × 106	2.37 × 1012	>0.99
	G3	3.50 × 105	2.34 × 1012	>0.99
With 100 mesh particles	G1	8.80 × 105	1.87 × 1012	>0.99
	G2	4.94 × 106	2.61 × 1012	>0.99
	G3	3.91 × 106	8.81 × 1012	>0.99
With 200 mesh particles	G1	1.72 × 106	2.13 × 1013	>0.99
	G2	1.01 × 107	1.57 × 1013	>0.99
	G3	1.21 × 107	4.03 × 1013	>0.99
No filling	G1	8.60 × 107	7.14 × 1013	>0.99
	G2	5.70 × 107	4.22 × 1013	>0.99
	G3	1.01 × 108	6.07 × 1013	>0.99

shows comparisons of the linear coefficient A and the nonlinear coefficient B. Coefficients A and B increase with a decrease in the particle size of the filling material, and the maximum coefficients are obtained under the condition without filling. Coefficient A is related to the viscous friction at the water and rock interface and represents the intrinsic permeability. It seems that a smaller particle of sand forms a denser pore structure, which makes the flow channel become narrower, and results in a lower flow capacity and permeability. Coefficient B describes the inertia effect due to an irreversible kinetic energy loss caused by flow acceleration or deceleration. The contact area of the fracture surface increases with a decrease in the particle size of the filling material, thus leading to an increment in the curvature of the flow path and producing an increase in the inertia effect during the flow process thereby ultimately, increasing Coefficient B.

As shown in Figure 7, the Reynolds number increases with an increase in water pressure. This indicates that the inertial force is more and more obvious than the viscous force in the process of increasing water pressure. This is because the flow velocity increases gradually with the increase in water pressure, and the inertial losses caused by high velocity result in a significant nonlinearity of flow. The critical point of the flow regime transformation from linear to nonlinear can be obtained by the calculation of the critical Reynolds number. For the unfilled condition, the range of the critical Reynolds number is 2.77~3.84, and the corresponding water pressure is 0.7~1.4 MPa. However, for the conditions of filling with quartz sand, the flow regime was nonlinear when the water pressure was 0.1 MPa. This is because the larger hydraulic aperture formed by filling with quartz sand (compared with the unfilled condition), resulted in a higher flow velocity under the same water pressure, and the inertial force caused by a high flow velocity makes the flow nonlinear under a low water pressure.

Analysis of the above-mentioned test results shows that the flow regime will change from linear to nonlinear under a high flow velocity in the fracture, and the variation in effective stress leads to a change in the hydraulic aperture, which has effects on the properties of nonlinear flow.

3. Hydro-Mechanical Coupled Model for Fractured Rock Mass

3.1. Governing Equations

A rock matrix is regarded as impermeable because of its extremely low permeability, and hydro-mechanical coupling only occurs in fractures. In this paper, the hydro-mechanical coupled model in a fractured rock mass has been discussed as follows:

Compression deformation of a fracture will occur due to normal stress, which leads to variation in the permeability with a change in the fracture’s aperture. In this paper, it is assumed that the normal stiffness of fractures is constant [67,68]. Under the action of normal stress, the fracture aperture $\mathrm{e}$ is calculated by the following formula:

$$\mathrm{e}={\mathrm{e}}_0+{\mathsf{\sigma }}_{\mathrm{n}}/{\mathrm{K}}_{\mathrm{n}}$$

(7)

where e₀ is the initial aperture, ${\mathsf{\sigma }}_{\mathrm{n}}$ is the normal stress of the fracture surface, the compressive stress is negative, and K_n is the normal stiffness of the fracture.

The intrinsic permeability of fracture K is calculated by the following formula [69,70]:

$$\mathrm{K}=\frac{\mathsf{\rho }\mathrm{g}{\mathrm{e}}^2}{12\mathsf{\mu }}$$

(8)

where $\mathsf{\rho }$ is the fluid density, $\mathrm{g}$ is the gravitational acceleration, and $\mathsf{\mu }$ is the dynamic viscosity coefficient of the fluid. It should be noted that Formula (8) is valid only when the fracture aperture e is greater than 0; when the aperture is less than 0, that is, when the fracture surface is embedded, the inherent permeability tends to 0.

Equation (7) is introduced into Equation (8) to obtain:

$$\mathrm{K}=\frac{\mathsf{\rho }\mathrm{g}{({\mathrm{e}}_0+{\mathsf{\sigma }}_{\mathrm{n}}/{\mathrm{K}}_{\mathrm{n}})}^2}{12\mathsf{\mu }}$$

(9)

To remain consistent with the first-order term of Darcy’s law, Equation (4) is often expressed as:

$$-\frac{\mathrm{d}\mathrm{P}}{\mathrm{d}\mathrm{X}}=\frac{\mathsf{\rho }\mathrm{g}}{\mathrm{K}}\mathrm{v}+\mathsf{\beta }\mathsf{\rho }{\mathrm{v}}^2$$

(10)

where P is the water pressure, v is the fluid velocity, and β is the non-Darcy coefficient. It is related to the permeability and porosity of fractured media and is inversely proportional to the power of permeability [71,72]. $\mathsf{\rho }$ is the fluid density.

$$\mathrm{v}=-\frac{1}{\left(1/\mathrm{K}+\frac{\mathsf{\beta }\mathrm{v}}{\mathrm{g}}\right)}\mathrm{J}$$

(11)

where J is the hydraulic gradient.

In the following equation, K_F is defined as the Forchheimer-type non-Darcy permeability:

$${\mathrm{K}}_{\mathrm{F}}=1/(1/\mathrm{K}+\mathsf{\beta }\mathrm{v}/\mathrm{g})$$

(12)

Equation (6) is introduced into Equation (9) to obtain the following:

$${\mathrm{K}}_{\mathrm{F}}=1/(12\mathsf{\mu }/\mathsf{\rho }\mathrm{g} {({\mathrm{e}}_0+{\mathsf{\sigma }}_{\mathrm{n}}/{\mathrm{K}}_{\mathrm{n}})}^2+\mathsf{\beta }\mathrm{v}/\mathrm{g})$$

(13)

The influence of the normal stress of a fracture changes the permeability of a fracture, which will lead to changes in the flow field and water pressure distribution in the fracture, and these changes have an effect on the stress field. The stress balance equation of a rock mass can be expressed by the principle of virtual work:

$${\displaystyle \underset{\mathrm{v}}{\mathsf{\int }}\mathsf{\delta }{\mathsf{\epsilon }}^{\mathrm{T}}}\mathrm{d}\mathsf{\sigma }\mathrm{d}\mathrm{V}-{\displaystyle \underset{\mathrm{v}}{\mathsf{\int }}\mathsf{\delta }{\mathrm{u}}^{\mathrm{T}}}\mathrm{d}\mathrm{f}\mathrm{d}\mathrm{V}-{\displaystyle \underset{\mathrm{S}}{\mathsf{\int }}\mathsf{\delta }{\mathrm{u}}^{\mathrm{T}}}\mathrm{d}\mathrm{t}\mathrm{d}\mathrm{S}=0$$

(14)

where $\mathrm{t}$ is the surface force, f is the physical strength, $\mathsf{\delta }\mathsf{\epsilon }$ and δu are the virtual displacement and virtual strain, respectively.

The constitutive model for the fracture is stated as follows:

$$\mathrm{d}{\mathsf{\sigma }}^{\prime }={\mathrm{D}}_{\mathrm{e}}(\mathrm{d}\mathsf{\epsilon }-\mathrm{d}{\mathsf{\epsilon }}_{\mathrm{s}})$$

(15)

where ${\mathrm{D}}_{\mathrm{e}}$ is the elastic matrix and dε_s is the normal strain of fracture caused by water pressure. The specific expression is stated as follows:

$$\mathrm{d}{\mathsf{\epsilon }}_{\mathrm{S}}=-\mathrm{m}\frac{\mathrm{d}\overline{\mathrm{p}}}{3{\mathrm{K}}_{\mathrm{s}}}$$

(16)

where $\mathrm{m}={\left[1,1,1,0,0,0\right]}^{\mathrm{T}}$, $\overline{\mathrm{p}}$ is the fluid pressure, and K_S is the compression modulus of the rock mass.

The software ABAQUS adopts the direct coupling method based on Biot’s consolidation theory, and the virtual work principle and effective stress principle are used to discretize the finite element on the basis of the stress–strain relationship and flow law. The coupling mathematical model is established [73] via:

$$\left[\begin{array}{cc}\overline{\mathrm{K}}& \mathrm{C}\\ \mathrm{E}& \mathrm{G}\end{array}\right]\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\left\{\begin{array}{c}\overline{\mathrm{u}}\\ {\overline{\mathrm{p}}}_{\mathrm{w}}\end{array}\right\}+\left[\begin{array}{cc}0& 0\\ 0& \mathrm{F}\end{array}\right]\left\{\begin{array}{c}\overline{\mathrm{u}}\\ {\overline{\mathrm{p}}}_{\mathrm{w}}\end{array}\right\}=\left\{\begin{array}{c}\frac{\mathrm{d}\overline{\mathrm{f}}}{\mathrm{d}\mathrm{t}}\\ \widehat{\mathrm{f}}\end{array}\right\}$$

(17)

where $\overline{\mathrm{u}}$ is the nodal displacement of the element, ${\overline{\mathrm{p}}}_{\mathrm{w}}$ is the nodal pore pressure of the element, $\overline{\mathrm{K}}$ is the stiffness matrix, C is the nodal force corresponding to the pore pressure of the node, E is the volume change corresponding to node displacement, $\mathrm{G}$ is the water storage matrix, $\mathrm{F}$ is the seepage matrix, $\overline{\mathrm{f}}$ is the nodal load, and $\widehat{\mathrm{f}}$ is the nodal flow.

The mechanism of a hydro-mechanical coupled system is shown in Figure 8a. In the actual process of fluid flow, on the one hand, due to the change of water pressure in the fracture, the system will cause a change in the effective stress on the fracture surface (Equation (15)). On the other hand, the change in effective stress will lead to a change in the fracture aperture (Equation (7)), and the permeability also change (Equation (13)); these changes will have an effect on the flow and distribution of water pressure. The solution process is based on ABAQUS (Equation (17)).

During the solution of ABAQUS, CPE6 element is used for the rock matrix, and CPE6MP element is used for the fracture as shown in Figure 8b. A subroutine is used to obtain the normal stress and flow velocity for each fracture element and to calculate the ultimate permeability coefficient ${\mathrm{K}}_{\mathrm{F}}$. In order to solve the coupled mathematical model, ABAQUS also needs to set corresponding definite solution conditions, such as the stress boundary, displacement boundary condition, and initial head boundary.

3.2. Fracture Network Generation for Rock Mass

In order to understand the flow behavior of a jointed rock mass, the fractured network should be built first, as it relates to the statistic distribution of fractures. In general, the characteristic fracture geometry parameters, including the inclination angle, spacing distribution, and trace length, follow certain statistical distributions in space. The process of generating the network and numerical mesh can be stated as follows:

(1) Assuming that the crack dip angle and length follow a normal distribution law, $\mathsf{\theta }$ represents the geometric parameters of the fracture angle and length, and the probability density function $\mathrm{f}(\mathsf{\theta })$ has the following form:

$$\mathrm{f}(\mathsf{\theta })=\frac{1}{\sqrt{2\pi {\mathsf{\sigma }}^2}}\mathrm{e}\mathrm{x}\mathrm{p}[-\frac{1}{2}{\left(\frac{\mathsf{\theta }-\mathsf{\mu }}{{\mathsf{\sigma }}^2}\right)}^2],(-\infty <\mathsf{\theta }<+\infty )$$

(18)

where $\mathsf{\mu }$ and $\mathsf{\sigma }$ are the expected value and standard variance of the parameters of the fracture network geometry, respectively.

(2) The geometric parameters of a single fracture applied in the DFN, such as the trace length $\mathrm{L}(\mathrm{i})$, the dip angle $\mathsf{\theta }(\mathrm{i})$, and the coordinates of the start point $({\mathrm{x}}_0(\mathrm{i}),{\mathrm{y}}_0(\mathrm{i}))$ are elected as the most significant input data. The coordinates of the end points are calculated with $\mathrm{L}(\mathrm{i})$, $\mathsf{\theta }(\mathrm{i})$, and $({\mathrm{x}}_0(\mathrm{i}),{\mathrm{y}}_0(\mathrm{i}))$.

(3) By copying a single fracture that has been established and shifting it by the distance of the desired fracture width, a fracture with thickness can be obtained, and the fracture aperture is the shifted distance.

(4) Several fractures are interwoven into the fracture network, but there are some isolated fractures and dead ends in the fracture network. Therefore, the isolated fractures and dead ends in the fracture network are removed to form a fully interconnected fracture network. According to the linear equation for a fracture element, all fracture elements can be selected and set up.

For generating the mesh of the random discrete fracture network model through ABAQUS, a triangular element is more suitable for regional division when multiple fractures are being considered. After mesh generation, matrix elements and fracture elements are distinguished by the geometric relationship between the linear equation, where the fractures are located, and the element nodes.

4. Numerical Results and Discussion

4.1. Experiment Validation

Figure 9 shows comparisons between the above experimental results of the relationship between the water pressure gradient and the flow rate with that of the numerical simulation. The results indicate high consistency for all cases. This shows that the nonlinear and hydraulic coupling effect of the fluid flow proved by the above experiments can be realized by numerical simulation.

A two crossed fractures model was established, and the intersection angle of the two fractures was 60°, as shown in Figure 10. The fracture aperture was set up to 1 mm when generating the models. These simulations include three cases of setting only one outlet and three cases of setting different combinations of two outlets. A comparison of the simulation results with the test results [74] is shown in Figure 11, where “E” represents the test results, “N” represents the numerical results, and the figure represents the number of the selected outlet. The results indicate high consistency for all cases.

A multiple crossed fractures model is shown in Figure 12. The length and width of this model are 20 cm and 10 cm, respectively. It is composed of five fracture sections, two inlets, and two outlets. Different fracture sections have different apertures, dip angles, and lengths, as shown in

Table 3. Basic information about the fracture section.

Fracture Section	Length (cm)	Aperture (mm)	Width (cm)
1	5.78	0.6	10
2	5.78	0.3	10
3	5.78	0.3	10
4	5.78	0.6	10
5	10	1	10

. Because the model has two inlets and two outlets, different combinations can be carried out, and the four working conditions shown in

Table 4. Simulated cases.

Cases	Inlet (s)	Outlet (s)	Remarks
Case 1	1	2	One inlet and one outlet
Case 2	1	1,2	One inlet and two outlets
Case 3	1,2	2	Two inlets and one outlet
Case 4	1,2	1,2	Two inlets and two outlets

are simulated.

As shown in Figure 13, “E” represents the test results and “N” represents the numerical results. Figure 13 shows a comparison between the numerical results and the corresponding test results [75], which again shows high consistency for all cases.

4.2. Flow Behaviours in DFNs

Nonlinear flow behavior exists not only in a single fracture or a small number of fractures but also in DFNs. Experimental and numerical research on a single fracture can also be applied to the complex fracture network. A fracture network is generated according to the method described in Section 3.2, as shown in Figure 14. The side length of the DFN model is 50 m. As the permeability of the rock matrix is too low, it is considered that the fluid only flows in the fractures. The horizontal in-situ stress ${\mathsf{\sigma }}_{\mathrm{x}}$ and vertical in-situ stress σ_y are applied to the model. The in-situ stress values are 1 MPa, 3 Mpa, 5 MPa, 7 MPa, and 10 MPa, respectively, and the lateral pressure coefficient is 1. The upper boundary of the model is the hydraulic boundary. The water pressure at the lower boundary is 0, and the left and right boundaries are impermeable boundaries. The range of the hydraulic gradient is 10⁻⁵~10⁰. The initial fracture aperture values considered are 1 mm, 1.5 mm, and 2 mm. The statistical distribution parameters of the two groups of fractures and the material parameter values of the matrix and fracture are shown in

Table 5. Distribution parameters of the two groups of fractures.

	Mean Value (Expectation)	Standard Variance
Trace length (m)	15	4
	15	2
Dip (°)	30	5
	120	7
Spacing	Random distribution

and

Table 6. Material parameters of the matrix and fractures.

	Elastic Modulus	Normal Stiffness	Cohesion	Friction Angle	Poisson’s Ratio	Density
Matrix	10 GPa	/	1.6 MPa	45°	0.3	2000 kg/m3
Fractures	/	40 MPa/mm	/	/	0.25	1500 kg/m3

, respectively. The material parameters are assigned to CPE6 element for the rock matrix and CPE6MP element for the fracture, respectively, before solving by ABAQUS.

Figure 15 shows that there is a nonlinear relationship between the hydraulic gradient and discharge. With an increase in the hydraulic gradient, the nonlinear phenomenon becomes more obvious, and this nonlinear relationship can be described well by the Forchheimer law. The numerical results show that the curvature of the curve increases with the hydraulic gradient, which indicates that the nonlinearity of Q and J becomes stronger under a higher flow velocity and the correlation coefficient R² > 0.99. Figure 15 shows that a larger in-situ stress induces smaller discharge under the same hydraulic gradient. Since the fracture would close under the influence of the normal stress, the larger the normal stress is, the smaller the equivalent hydraulic aperture will be, which induces the lower permeability. Wang et al. [51] uses Boltzmann’s simulation method to simulate the nonlinear flow of fracture surfaces with different roughnesses. This nonlinear characteristic is consistent with the simulation results of this study; this view was also confirmed by Liu et al. [50]. In addition, the influence of stress on aperture was considered in this paper.

The relationship between the transmissivity (the ratio of the discharge to hydraulic gradient) and hydraulic gradient of the model is shown in Figure 16. The curve can be roughly divided into three stages. In the first stage, when the hydraulic gradient is less than 10⁻³, the flow velocity is low in a fracture, and the effect of inertia can be ignored. The transmissivity is constant, which indicates that the fluid flow is linear and that the cubic law is applicable. In the second stage, when the hydraulic gradient is less than 10⁻² but greater than 10⁻³, the transmissivity changes slowly with the hydraulic gradient, indicating that there is a weak inertial region in this stage. In the third stage, when the hydraulic gradient is greater than 10⁻², the flow velocity is high, and the effect of inertia plays a leading role, which cannot be ignored during this stage. The transmissivity decreases significantly with the hydraulic gradient, which indicates that there is a strong inertial region in which the cubic law is inapplicable. Figure 16 also shows that with an increase in in-situ stress, the permeability of the fracture network decreases because the larger normal stress will lead to a smaller hydraulic aperture. However, due to only a small change in aperture under the normal stress, there is no change in the magnitude of the critical hydraulic gradient (it remains between approximately 10⁻³ and 10⁻²). It was also proven that three regions from linear to nonlinear exist with an increase in the hydraulic gradient in the numerical study carried out by Liu et al. [50] and Yin et al. [57], which is similar to the results of this paper.

Figure 17 shows that with increasing confining pressure, both the linear coefficient A and nonlinear coefficient B increase. However, with an increasing fracture aperture, both the linear coefficient and nonlinear coefficient decrease. The fracture fits more closely due to the increase in confining pressure; larger confining stress leads to greater closure of a fracture, and the smaller the aperture is, the smaller the inherent permeability will be, which leads to an increase in Coefficient A. The close fitting of the fracture surface increases the difficulty of water flow, and the increase in contact area leads to the increase in water flow curvature, which increases Coefficient B. Similarly, an increase in the initial fracture aperture weakens this effect. The nonlinear coefficient increases more than the linear coefficient when the confining pressure increases or the fracture aperture decreases, showing that the nonlinear term is more sensitive to changes in the confining pressure and aperture than the linear term. Through the nonlinear flow experiment in a single fracture, carried out by Chen et al. [53], the influence of confining pressure on nonlinear flow characteristics was studied. The results show that Coefficients A and B gradually increase with an increase in confining pressure, which is consistent with the study results obtained by the numerical simulation of a fracture network in this paper.

Figure 18 shows the local velocity contour and vector diagram of the fracture intersection area. Figure 18a indicates that the existence of the fracture intersection affects the continuous distribution of velocity in the fracture. As shown in Figure 18b, the direction and velocity of the flow at the fracture intersection will be determined according to the length and hydraulic aperture of fractures and the pressure gradient, and its direction is no longer consistent with the direction of the fracture. A deviation from the direction will increase the energy loss caused by water flow and enhance its inertia, which will accelerate the occurrence of nonlinear flow.

4.3. Case Study for Underground Engineering

Considering the influence of nonlinear flow on the underground excavation process, to accurately predict the flow discharge in the excavation process, a 2D network model of fractured rock masses with tunnel excavation is shown in Figure 19. The statistical distribution parameters of the two groups of fractures are shown in

Table 7. Distribution parameters of the two groups of fractures.

	Mean Value (Expectation)	Standard Variance
Trace length (m)	20	4
	18	2
Dip (°)	30	5
	120	7
Spacing	Random distribution

. The burial depth of the tunnel is 500 m, the boundary length of the model is 200 m, and the diameter of the tunnel is 16 m. The in-situ stress of the calculation model is determined according to the rock density and buried depth. The horizontal in-situ stress σ_x and vertical in-situ stress σ_y are 10 MPa, and the lateral pressure coefficient is 1.

Table 8. Material parameters of the rock matrix and fractures.

	Elastic Modulus	Normal Stiffness	Cohesion	Friction Angle	Poisson’s Ratio	Density
Matrix	10 GPa	/	1.5 MPa	42°	0.27	2150 kg/m3
Fractures	/	40 MPa/mm	/	/	0.25	1850 kg/m3

lists the material parameter values of the surrounding rock matrix and fracture. The material parameters are assigned to CPE6 element for the rock matrix and CPE6MP element for a fracture, respectively, before solving by ABAQUS. As shown in

Table 9. Parameter values and calculation results of cases.

Fracture Aperture (mm)	Water Pressure (MPa)	Flow (m3/s)	Reynolds Number	Critical Reynolds Number
0.1	0.1	9.77 × 10−5	98	347
	0.3	2.79 × 10−4	279
	0.5	4.46 × 10−4	446
	1.0	8.07 × 10−4	807
0.5	0.1	6.26 × 10−4	626	1288
	0.3	1.86 × 10−3	1863
	0.5	3.10 × 10−3	3100
	1.0	5.19 × 10−3	5191
1.0	0.1	2.39 × 10−3	2386	2072
	0.3	5.96 × 10−3	5960
	0.5	8.95 × 10−3	8950
	1.0	1.47 × 10−2	14,740

, the fracture apertures are set to 0.1 mm, 0.5 mm, and 1 mm. The upper and lower boundaries of the model are set to have a water pressure of 0.1–1 MPa. The water pressure of the excavation face is set to zero, and the left and right boundaries are impermeable. When the water pressure increases from 0.1 MPa to 1 MPa, the flow rates, calculated based on the developed model for fracture apertures of 0.1 mm, 0.5 mm, and 1 mm, are in the ranges of 9.77 × 10⁻⁵~8.07 $\times$ 10⁻⁴ m³/s, 6.26 × 10⁻⁴~5.19 $\times$ 10⁻³ m³/s, and 2.39 $\times$ 10⁻³~1.47 × 10⁻² m³/s, respectively. According to Figure 20, higher water pressure and a larger aperture lead to a greater flow rate. When the fracture aperture is 0.1 mm and 0.5 mm, the flow increases almost linearly with an increase in water pressure. However, the flow no longer increases linearly with an increase in water pressure when the aperture is 1 mm, and the increase rate decreases gradually.

As Figure 21 shows, when the water pressure increases from 0.1 MPa to 1 MPa, the flow rates, calculated based on the developed model for fracture apertures of 0.1 mm, 0.5 mm, and 1 mm, are in the ranges of 98–807, 626–5191, and 2386–14,740, respectively. The Reynolds number increases gradually with an increase in water pressure. When the hydraulic aperture are 0.1 mm, 0.5 mm, and 1 mm, the water pressure corresponding to the critical Reynolds number is about 0.38 MPa, 0.2 MPa, and 0.09 MPa, respectively, which shows that the smaller the hydraulic aperture, the higher the water pressure needed to make the flow regime change from linear to nonlinear. This elaborates that the larger the aperture is, the higher the flow velocity will be. The inertial force caused by high velocity plays a significant role in nonlinear flow, which results in an increased possibility that the nonlinear flow in a fractured rock mass will occur under lower water pressure. In contrast, when the aperture is small, the velocity is very low under the same water pressure conditions, so the inertial force can be neglected, and the flow obeys Darcy’s law.

A comparison of the pore pressure in the fractures before and after excavation and support is shown in Figure 22. After tunnel excavation, the surroundings of the tunnel are permeable, and thus the water pressure around the tunnel is zero, corresponding to the “p = 0” stage in Figure 22. There is a gradient distribution of water pressure from the model boundary to the tunnel periphery. The “T = 15 s” stage in Figure 22 shows the water pressure distribution during the application of the lining. It indicates that the water pressure around the tunnel periphery increases from zero gradually during the process of the application of the lining. Along with the support being applied, the permeability of the fractured rock mass is relatively high, even if the aperture is 0.1 mm, and the large boundary water pressure makes the water pressure around the support increase quickly, resulting in a not very obvious difference as compared with the condition of the 1-mm aperture. After the lining is installed, the surrounding of the tunnel is impermeable, and the water pressure of boundary is equal to the tunnel periphery, finally corresponding to the “Q = 0” stage in Figure 22, and the distributions of the water pressures with apertures of 0.1 mm and 1 mm are consistent.

5. Conclusions

In this paper, the nonlinear flow behavior of high-velocity fluid in fractured rock masses is further studied through laboratory experiments, theoretical solutions, and numerical simulations. The following can be concluded:

The flow experiments of a single fracture indicate that a high water pressure and fluid velocity in the fracture will make the flow change to nonlinear under the action of inertial force. The Forchheimer law could describe the relationship between the hydraulic gradient and flow well, and a greater water pressure leads to a stronger nonlinearity of the flow. Further, the linear and nonlinear coefficients of the Forchheimer law increase with a decrease in the particle size of the filling material in fractures, and the maximum of the coefficients are obtained under the unfilled condition.

Furthermore, the classical Forchheimer law is developed to consider the influence of stress on the normal closure of fractures; the developed model is suitable not only for single fractures but also for discrete fracture networks. The numerical results show that normal stress will induce closure of a fracture and result in a smaller hydraulic aperture; these changes lead to a lower inherent permeability and result in an increase in the linear coefficient and nonlinear coefficient. The numerical results from the fractured rock masses show that the ratio of the flow to the hydraulic gradient will change the flow from linear to weak nonlinearity and, finally, to strong nonlinearity with an increase in the hydraulic gradient. The complexity of the fracture network and the existence of fracture intersections make the nonlinear flow occur more more.

A case study on the excavation of tunnel engineering was carried out through numerical simulation. The results show that the Reynolds number increases gradually with an increase in water pressure and that nonlinear flow is more likely to occur under lower water pressure for a larger hydraulic aperture. Therefore, in rock engineering, Darcy’s law is no longer useful under conditions of high water pressure and high velocity, and a nonlinear relationship that considers the influence of in-situ stress is needed to describe the flow. With this approach, the flow conditions during underground excavation can be predicted more accurately.

In this paper, the influence of the change in aperture caused by normal stress on the nonlinear flow of a fractured rock mass is systematically studied through experiment, theory, and numerical simulation. Further experimental and numerical studies will be carried out to investigate the influence of shear on the nonlinear flow characteristics of fractured rock masses.