Introduction

Understanding the hydraulic characteristics of fractured rock masses plays an important role in oil exploitation, hydropower engineering, pollutant control engineering, etc. (Qian et al. 2011; Wang et al. 2015; Chen et al. 2016; Qiao et al. 2017; Li et al. 2020). The fracture network in a given rock mass is very complex, and its seepage characteristics are directly related to numerous factors. In fractured rock masses, the permeability of the intact rock masses is usually very low, and the fracture network provides the main channels of fluid flow (Su et al. 1999; Cherubini et al. 2012; Wang et al. 2013; Yang et al. 2014). It is reasonable to assume that the fracture aperture and connection form are related to the flow characteristics of the fluid occurring in the fracture. In related research, appropriate analysis of fracture network flow at the laboratory scale constitutes the basis of understanding flow through a real complex fracture rock mass network (Baghbanan and Jing, 2007; Takahashi et al. 2013; Wang et al. 2014; Shang et al. 2019).

The basic forms of fracture networks include single fractures and cross fractures. Different scholars have performed much research on single-fracture seepage through theoretical analysis, numerical calculation, and experimental exploration (Lomize, 1951; Louis 1969; Skjetne et al. 1999; Berkowitz 2002; Scesi and Gattinoni 2007). In regard to fracture seepage, when the Reynolds number is low, the flow and pressure gradient generally conform to the linear cubic law; with increasing Reynolds number, the flow rate and pressure gradient gradually deviate from the linear relationship, and the resultant nonlinear characteristics can be described with the Forchheimer equation (Whitaker 1996; Konzuk and Kueper 2004; Qian et al. 2011; Kong et al. 2018). Su (1997) and Hu et al. (2005) studied the local flow state in cross fractures. According to the fluid momentum equation and considering the velocity head, the local head loss equation of a cross fracture with an undetermined coefficient was derived. Johnson et al. (2006) observed the morphology of dyed fluid flow through rough cross-fracture model tests and concluded that rough cross fractures greatly increase the mixing of fluid at fracture intersections. Zhu et al. (2013) pointed out that due to the intersection and deflection of fluids in cross fractures, the actual pressure drop at both ends of the fracture, which was larger than the theoretical pressure drop, was defined as the excess pressure drop loss, and the excess pressure drop loss coefficient ζ was proposed to quantify the degree of pressure loss.

At present, discrete fracture networks (DFNs) are widely applied in the study of fracture network seepage and attain a good application effect in solving large-scale engineering problems, but research on fracture network seepage is relatively limited at the mesoscale (Kulatilake and Panda, 2000; Chen et al. 2008; Klimczak et al. 2010; Wu and Kulatilake 2012; Zhang et al. 2012). Through the study of the flow distribution in a fracture network, it has been found that fluid flow mainly occurs in long fractures parallel to the direction of fluid flow or hydraulic gradient, especially in those fractures intersecting with the entrance and exit boundaries of the model, thus forming the dominant flow path in fracture network seepage (Liu et al. 2015; Yu et al. 2017). Research on the seepage characteristics of fracture networks with regular geometries, represented by basalt columnar jointed rock masses, has become a new research topic of heightened interest. Researchers have mainly studied this topic through the physical model test method. The corresponding theoretical model has seldom considered the nonlinear seepage characteristics of cross fractures, so it does not suitably quantify the loss value (Zou et al. 2013; Niu et al. 2020). Based on the research of the above scholars, at present, scholars do not fully understand the phenomenon of energy loss caused by head loss, have not formed a theoretical model that can describe the head loss in complex fracture network, and lack of understanding of the meso mechanism of head loss effect.

To further study the influence of the local head loss on the seepage characteristics of the fracture network, the fracture network has been simulated as an electric circuit network (Kong et al. 2018). Considering the local pressure drop loss effect at the intersection of fractures, an equivalent local loss resistance model is established based on the equivalent seepage resistance model. Moreover, aiming at the common cross-section forms of columnar joints, physical models of quadrilateral and hexagonal fracture networks are established, and seepage tests under different pressure drop conditions are carried out based on these physical models. Finally, the test results are compared to theoretical calculation results. With the use of general-purpose CFD software FLUENT, the microscale flow state in the fracture network is analyzed, and streamline and velocity vector characteristics of the fracture network are obtained. Hence, a comprehensive analysis of the seepage characteristics of a complex fracture network considering local loss is realized.

Methodology

The research method is divided into three parts, and the technical roadmap of this study is shown in Fig. 1.

Fig. 1
figure 1

The technology roadmap of the work

Full size image

Equivalent circuit characteristics of the fracture network

The fracture network is composed of a limited number of single fractures, and the most basic connection modes are the two modes shown in Fig. 2. The seepage law of smooth single fracture conforms to the cubic law, in which the volumetric flow q through the fracture can be expressed as:

$$q=\frac{\rho gw{e}^{3}}{12\mu }J$$
(1)

where ρ is the density of the fluid, g is the acceleration of gravity, w is the thickness of the fracture, e is the opening of the fracture, μ is the dynamic viscosity coefficient of the fluid, and J is the hydraulic gradient along the flow direction.

Fig. 2
figure 2

Different connection modes of fractures: a parallel connection mode of fractures, b series connection mode of fractures

Full size image

The parallel connection mode of fractures can be simplified, as shown in Fig. 2a, and the total fracture flow is equal to the sum of the flow through the two connected fractures:

$${q}_{J}={q}_{1}+{q}_{2}=\frac{w{e}_{1}^{3}}{12\mu {C}_{1}}\frac{\Delta {P}_{1}}{\Delta L}\text{+}\frac{w{e}_{2}^{3}}{12\mu {C}_{2}}\frac{\Delta {P}_{2}}{\Delta L}$$
(2)

where qJ is the total fracture flow, q1 and q2 are the flow amounts through the two fractures, ΔP1 and ΔP2 are the pressure drops in the two fractures, C1 and C2 are the roughness correction coefficients of the two fractures, ΔL is the lengths of the fractures, e1 and e2 are the widths of the two fractures, and w is the thickness of the fracture.

Correspondingly, the series connection mode of fractures is shown in Fig. 2b, and the total pressure drop of the connected fractures is equal to the sum of the pressure drop of each fracture:

$$\Delta P=\Delta {P}_{3}\text{+}\Delta {P}_{4}=\frac{12\mu {C}_{3}\Delta {L}_{1}}{w{e}_{3}^{3}}{q}_{3}\text{+}\frac{12\mu {C}_{4}\Delta {L}_{2}}{w{e}_{4}^{3}}{q}_{4}$$
(3)

Comparing Eqs. (2) and (3), it is observed that a common \({12}\mu C\Delta L\text{/}w{e}^{3}\) term occurs in both equations. Tao and Liu (2012) defined this term as the equivalent seepage resistance Rf, so the seepage characteristics of a given fracture network are equivalent to the current characteristics of a circuit. Zhu et al. (2013) found that the calculated value of the total pressure drop attributed to an abrupt change in the fracture width within a fracture network is smaller than the test value and defined the loss difference as the excess water pressure drop loss. This phenomenon is caused by the following two factors: one factor is the change in velocity due to the abrupt change in fracture width, and the other factor is the change in the fluid flow state due to the abrupt change in fracture width. The loss coefficient of the excess water pressure drop ζ is defined as:

$$\zeta =\frac{\Delta {P}_{c}}{\Delta P}\le {1}$$
(4)

where ΔP is the measured water pressure drop and ΔPc is the calculated water pressure drop. The dimensionless expression of the water pressure drop ΔP is Δp. Substituting this dimensionless expression in Eq. (3), the following is obtained:

$$\zeta =\frac{{24}{e}_{3}^{2}L}{\Delta p\mathrm{Re}}(\frac{1}{{e}_{3}^{3}}+\frac{1}{{e}_{4}^{3}})$$
(5)
$$\mathrm{Re}=\rho vL/\mu$$
(6)

where ρ is the fluid density and μ is the dynamic viscosity coefficient, Re is the Reynolds number and v is the characteristic velocity of the flow field.

There are two types of fracture connections: gap width contraction and gap width expansion. Along the flow direction, the excess water pressure drop under the condition of gap width expansion can be ignored, and the relationship between Δp and Re is as follows (Zhu et al. 2013):

$$\Delta p=0.6+14.14/[\mathrm{Re}({e}_{1}/2L)],\mathrm{Re}\le \mathrm{Re}\text{c)}$$
(7)

where Rec is the critical Reynolds number. In addition, the dip angle is also an inherent property of fractures, and experimental research has shown that in most cases, the loss attributed to the excess water pressure drop caused by the fracture dip angle can be expressed based on the increase in fracture flow distance.

Establishment of the E-loss model of the fracture network

The most basic form of the mixed connection mode of a single-inlet fracture network is the single-inlet series parallel hybrid connection mode, as shown in Fig. 3.

Fig. 3
figure 3

Single-inlet series parallel hybrid connection mode

Full size image

Under this connection mode, without considering the loss attributed to the excess water pressure drop at the contraction position of the gap width, the total flow qJ of the fracture is as follows:

$${q}_{J}=\Delta P/{R}_{J}=\Delta P/(\frac{12\mu {C}_{2}{C}_{3}L}{w{e}_{3}^{3}{C}_{2}\text{+}w{e}_{2}^{3}{C}_{3}}+\frac{12\mu {C}_{1}L}{w{e}_{1}^{3}})$$
(8)

where RJ is the total seepage resistance of the fracture.

It is difficult to consider the head loss based on the loss coefficient of the excess water pressure drop. Therefore, in previous research work, scholars have applied a simplified method to consider only the water pressure loss along the dominant seepage path. At e2 <  < e3, the simplified method is more reasonable, but when e2 approaches e3, the simplified method is not applicable (Sang et al. 2016).

To consider the head loss in the fracture network under the different fracture modes more accurately, the loss is defined as the equivalent local loss resistance Rx. As shown in Fig. 2b, the equivalent local loss resistance Rx can be expressed as:

$${R}_{x}\text{=}{R}_{J}-({R}_{f3}+{R}_{f4})\text{=}\frac{\Delta P}{q}-\frac{\Delta {P}_{c}}{q}=\frac{1-\zeta }{\zeta }({R}_{f3}+{R}_{f4})$$
(9)

where Rf3 and Rf4 denote the seepage resistance values of the two fractures connected in series. Substituting Eq. (9) into Eq. (7), Rx can be expressed as:

$${R}_{x}\text{=}(\frac{0.6\mathrm{Re}({e}_{3}/L)+28.28}{24[1+{({e}_{3}/{e}_{4})}^{3}]}-1)({R}_{f3}+{R}_{f4})$$
(10)

Based on the above equation, the key to model application to mixed joint fractures, as shown in Fig. 3, is to determine the Reynolds number of each fracture. To facilitate analysis, the ratio of the Reynolds number of the branch fractures (fractures ② and ③) to that of the main fractures (fracture ①) is defined as the Reynolds number distribution coefficient α:

$${\alpha }_{i}\text{=}\frac{{\mathrm{Re}}_{i}}{\mathrm{Re}}$$
(11)

Combining Eqs. (9), (10), and (11), the equivalent local loss resistance Rx2 of the fracture branch ② and the equivalent local loss resistance Rx3 of the fracture branch ③ can be obtained as follows:

$${R}_{x2}=(\frac{0.6{e}_{2}\mathrm{Re}/L[{({e}_{3}/{e}_{2})}^{3}+1]+28.28}{24[1+{({e}_{2}/{e}_{1})}^{3}]}-1)({R}_{f1}+{R}_{f2})$$
(12)
$${R}_{x3}=(\frac{0.6{e}_{3}\mathrm{Re}/L[{({e}_{2}/{e}_{3})}^{3}+1]+28.28}{24[1+{({e}_{3}/{e}_{1})}^{3}]}-1)({R}_{f1}+{R}_{f{3}})$$
(13)

The total flow through the fracture network and the flow through each branch can be expressed as follows:

$${q}_{J}=\Delta P/{R}_{J}=\Delta P/({R}_{f1}+\frac{({R}_{x2}+{R}_{f2})({R}_{x3}+{R}_{f3})}{{R}_{x2}+{R}_{f2}+{R}_{x3}+{R}_{f3}})$$
(14)
$${q}_{2}\text{=}{q}_{J}\cdot {\alpha }_{2},{q}_{3}\text{=}{q}_{J}\cdot {\alpha }_{3}$$
(15)

The most basic form of the mixed-connection mode of a fracture network with dual inlets is the cross-shaped fracture formed by the intersection of two fractures, as shown in Fig. 4.

Fig. 4
figure 4

Dual-inlet series parallel hybrid connection mode

Full size image

Tian (1986) divided the water flow in a cross fracture into three streams through dye tracer tests: the α stream from the narrow fracture passing through the intersection to the wide fracture; the β stream from the wide fracture passing through the intersection to the narrow fracture, and the third stream from the wide fracture passing through the intersection to the wide fracture. Among these three flows, only the β flow passes through the contraction joint, so only the equivalent local loss resistance caused by the β flow should be considered. According to the E-loss model, the total discharge of cross fractures is as follows:

$${q}_{J}=P/(R{^{\prime}}_{f}+R{^{\prime\prime} }_{f})=\frac{({R}_{f1}+{R}_{f2})({R}_{f1}+{R}_{f2}+{R}_{x})}{2{R}_{f1}\cdot {R}_{f2}({R}_{f1}+{R}_{f2}+{R}_{x})+{R}_{f2}^{2}\cdot {R}_{x}}P$$
(16)

In the above equation, \({{R}^{^{\prime}}}_{f}\text{ and }{{R}^{^{\prime\prime} }}_{f}\) are the equivalent seepage resistance values of the left and right sides, respectively, of the o point, and Rx is the equivalent local loss resistance caused by the β flow.

Through the above analysis, with the E-loss model, the seepage problem of a complex parallel connection fracture network can be decomposed into an equivalent circuit problem including multiple equivalent seepage resistance and equivalent local loss resistance values, and the flow distribution in each fracture of the fracture network can be obtained by an iterative method. The physical meaning is clear, and the calculation accuracy is high.

Scheme and method of the physical model test

Combined with the research on the seepage characteristics of fracture networks in columnar jointed rock masses, physical models of quadrilateral fracture networks and hexagonal fracture networks are established to study the local head loss effect in complex fracture networks. In this paper, quadrilateral and hexagonal fracture networks are designed and manufactured with smooth plexiglass as the main material, as shown in Fig. 5. The whole test device consists of three parts: a fixed water head device (responsible for providing a controllable and stable fixed water head), a fracture network model (the main test area), and a pressure measuring device (responsible for measuring the water head at each outlet of the fracture network).

Fig. 5
figure 5

Fracture network model: a quadrilateral fracture network model, b hexagonal fracture network model

Full size image

The physical model test is divided into four steps:

  • : Connect the reserved holes of the fracture network model to the corresponding positions with rubber pipes and assess the sealing effect of the whole model.

  • : Lift the constant-head water tank to a suitable height, rotate the ball valve to adjust the water head, ensure that there are no bubbles in the model, and record the water head at the inlet and outlet ends after the flow rate has stabilized.

  • : Measure the outflow amount at the three outlets and determine the average value of three measurements for each group.

  • : Gradually raise the constant-head device through the automatic lifting device and repeat the second and third steps to obtain the flow distribution at each outlet of the fracture network under different pressure drops.

Establishment of the numerical model

Computational fluid dynamics (CFD) has been widely applied to calculate the flow around single or multiple pipes within a large Reynolds number range (Lam et al. 2008). CFD software FLUENT based on the finite volume method is used to solve the corresponding Navier Stokes equations. The mass conservation equation or continuity equation in the fluid domain without a confluence source is as follows (Zhang et al. 2021):

$$\frac{\partial \rho }{\partial t}+\nabla (\rho \overrightarrow{v})=0$$
(17)

where ρ is the density of the liquid, t is the time, and \(\overrightarrow{v}\) is the velocity vector.

The law of momentum conservation represents the change in force in the fluid domain, and the momentum equation is expressed as follows:

$$\frac{\partial }{\partial t}(\rho \overrightarrow{v})+\nabla \cdot (\rho \overrightarrow{v}\overrightarrow{v})=-\nabla p+\nabla \cdot (\overline{\overline{\tau } })+\rho \overrightarrow{g}+\overrightarrow{F}$$
(18)

where p is the static pressure, \(\overline{\overline{\tau } }\) is the stress tensor, and \(\rho \overrightarrow{g}\) and \(\overrightarrow{F}\) are the gravitational body force and external body force, respectively.

In this paper, a three-dimensional model is established according to the actual model size. Three-dimensional modeling diagrams of the fluid domain of the above two fracture networks are shown in Fig. 6. The model is subsequently meshed. Due to the complexity of the model, an unstructured mesh is adopted, which mainly comprises tetrahedra, thus accelerating the convergence process of the solver. The two fracture network models are composed of 450,000 and 400,000 elements. After modeling, they are imported into the Fluent solver for calculation.

Fig. 6
figure 6

Three-dimensional models: a quadrilateral fracture network model, b hexagonal fracture network model

Full size image

The k-ɛ model is chosen as the turbulence model in the solver. The k-ɛ model was first put forward in the 1970s. It is a semiempirical equation established by W.P. Jones and B.K. Lauder considering experimental phenomena. The k-ɛ model is a two-equation turbulence model. The advantage of this model is that it improves the mixing length model and avoids the algebraic expression of the turbulent length in complex flow, so it achieves a good calculation effect and good stability and economy. In the pressure velocity coupling method, the pressure implicit with splitting of operators (PISO) algorithm, which yields obvious advantages in solving transient problems, is adopted. The method performs two additional corrections, namely, adjacency correction and skew correction, which makes the calculation process more efficient. The gradient term is based on the least-square method of the elemental volume, and the pressure interpolation scheme is based on the volume force weighting scheme. The wall adopts the assumption of fixed nonslip boundary. In terms of water head, it is set as a fixed pressure head according to the test conditions, and the outlet boundary is set as atmospheric pressure boundary.

Results and discussion

Experimental results and analysis of the quadrilateral fracture network

The outlet flow (qa, qb, and qc) at the three outlets (a, b, and c, respectively) of the quadrilateral fracture network under different pressure drop conditions and the total discharge qJ are listed in Table 1.

Table 1 Test results of the outlet flow of the quadrilateral fracture network
Full size table

Based on the experimental phenomena, it is observed that with increasing water pressure, the Reynolds number of the flow through each fracture gradually increases, and the main flow form in the fracture network changes from laminar flow to turbulent flow. Under a low pressure, the flow in the system mainly occurs in the stable laminar state, and the flow rate and total flow rate at each outlet are approximately linear with the water pressure drop. With increasing water pressure, the flow rate and water pressure drop attain a nonlinear corresponding relationship. In addition, due to the different water carrying capacities of each fracture in the fracture network system, the corresponding water pressure drop associated with each fracture flow regime is different, and the flow regime at each outlet changes to turbulent flow with decreasing gap width.

With increasing water pressure, the proportion of flow at outlet a slightly increases, the proportion of flow at outlet b notably decreases, and the proportion of flow at outlet c increases significantly. The evolution of the flow rate can be divided into the following three stages: ① at the laminar flow stage of the fracture network, when the pressure drop is small (smaller than 250 Pa), all single fractures occur in the laminar flow state, the seepage characteristics of the fracture network are only affected by the head loss at each node, and the flow rate at each fracture outlet remains almost constant. ② At the turbulent development stage of the fracture network, with increasing water pressure drop, the flow state in the fracture network part with a larger gap width first develops to turbulent flow, and the resistance of the fluid further increases. However, the flow state in the fractures with smaller gap widths is laminar flow, which leads to a decrease in the proportion of flow in the fractures with larger gap widths. The corresponding test results indicate that the proportion of fracture flow at outlet c greatly increases. ③ In the turbulent phase of the fracture network, when the water pressure drop is large enough (larger than 613 Pa), the main flow through the fractures of the fracture network enters the turbulent state. Subsequently, the fluid in each fracture is affected by turbulence, and the resistance increases. The further diversion effect caused by the turbulence in the large-width fractures weakens, and the proportion of flow at each outlet fracture tends to remain stable.

The numbered quadrilateral fracture network model is shown in Fig. 7a, and the results of the dyed tracer test are shown in Fig. 7b. The calculation results of the water flow through each fracture with the E-loss model proposed above are listed in Table 2.

Fig. 7
figure 7

Experimental results of the quadrilateral fracture network: a fracture network number, b results of the dyed tracer test

Full size image
Table 2 Flow calculation of the quadrilateral fracture network
Full size table

The tracer test reveals the water flow direction in the fracture network, and it is observed that the fluid in fracture 2 flows into fractures 7 and 9 through fractures 5 and 6. The fluid in fractures 7 and 9 then flows into fracture 14 through fractures 11 and 12, respectively. Moreover, the water velocity in transverse fractures 5, 6, and 11 is lower than that in longitudinal fractures 1, 2, 7, and 8. Table 2 provides the flow rate in each fracture calculated with the E-loss model. Based on the calculation results, it is found that the model reflects the results of the tracer test and verifies the rationality of the basic assumptions of the model.

Furthermore, the experimental results of the total discharge of the quadrilateral fracture network under the different pressure drop conditions, the calculation results of the local cubic law and the calculation results of the E-loss model are comprehensively shown in Fig. 8.

Fig. 8
figure 8

Comparison diagram of the test results and theoretical calculation results

Full size image

Figure 8 shows that when the water pressure drop Δp is small, the outlet flow calculated with the local cubic law is larger than the test value because the excessive water pressure drop loss at the fracture intersections is not considered, and the flow calculated with the method proposed in this paper is closer to the test data, thus verifying the applicability of the E-loss model. With increasing fracture water pressure drop, turbulent flow occurs in some fractures of the fracture network, but the basic assumption of the E-loss model is that fracture flow conforms to linear Darcy flow. Therefore, the calculation results based on this model gradually deviate from the test data, but they are still closer to the test values than those obtained with the cubic law, which achieves a certain correction effect. The process can be observed more intuitively with the total fracture discharge diagram. When the water pressure drop is larger than 613 Pa, the nonlinear characteristics of the overall fracture discharge rapidly increase. The boundary is defined as the laminar turbulent boundary (L–T line). When the pressure drop Δp is located below the L–T line, the experimental and theoretical values exhibit a higher fitting value. When the Reynolds number in each fracture of the fracture network continues to rise, the resultant turbulent flow exhibits nonlinear characteristics, and the calculated values of the seepage model based on the local cubic law deviate, so the turbulent flow theory should be considered to describe the deep flow characteristics of the fracture network, i.e., the characteristics above the L–T line.

By observing the relationship between the flow rate and the water pressure drop at each outlet, it is found that when the water pressure drop exceeds 250 Pa and less than 600 Pa, the actual flow rate at outlets a and b is lower than the theoretical calculation results based on the E-loss model, However, the flow value at outlet c even exceeding the calculation value obtained with the local cubic law, which is further verified: when the water pressure drop is larger than 250 Pa and some turbulent fracture fluid flow occurs, the resistance of the fluid in the turbulent region increases, and the fluid in the large-gap width fractures is further distributed into the small-gap width fractures. Therefore, the flow at outlet c increases, and the flow rate is higher than the calculated value obtained with the cubic law. However, with increasing pressure drop, when the water pressure drop is larger than 600 Pa, the fluid flow in the small-gap width fractures gradually transitions into turbulent flow, and the influence of turbulence effect in large-gap width fractures on the flow of small-gap width fractures is reduced, which results in a lower flow rate at outlet c than that calculated with the cubic law but still higher than that calculated with the E-loss model.

Above the L–T line, with increasing water pressure drop, the flow in each fracture of the fracture network gradually transitions to the turbulent flow state. With the use of the research method of Qian (Qian et al. 2011), the Forchheimer equation is adopted to fit the outlet flow and total flow. The fitting results reveal that the determination coefficient R2 of the discharge at each outlet and the total discharge is higher than 0.99, which suggests that the seepage characteristics of the fracture network can be described with the Forchheimer equation under turbulent flow conditions.

Experimental results and analysis of the hexagonal fracture network

The same analysis method is used to analyze the experimental results of the hexagonal fracture network, and the outlet flow (qd, qe, and qf) at the three outlets (d, e, and f, respectively) of the hexagonal fracture network under the different pressure drop conditions and the total discharge qJ are listed in Table 3.

Table 3 Test results of outlet flow of hexagonal fracture network
Full size table

The numbered hexagonal fracture network model is shown in Fig. 9a, and the results of the dyed tracer test are shown in Fig. 9b. The calculation results of fracture water flow by using the E-loss model proposed above are provided in Table 4.

Fig. 9
figure 9

Experimental results of the hexagonal fracture network: a fracture network number, b results of the dyed tracer test

Full size image
Table 4 Flow calculation results of the hexagonal fracture network
Full size table

According to the test data in Table 4, the flow evolution in the hexagonal fracture network is similar to that in the quadrilateral fracture network: the flow rate at each outlet is different, and the flow rate in the fractures with a larger gap width is higher. With increasing water pressure drop, the flow rate at the three outlets changes, the flow rate at outlet d slightly fluctuates, the flow rate at outlet d fluctuates slightly, the flow rate at outlet e decreases notably, and the flow rate at outlet f significantly increases. This also demonstrates that there are three stages in the hexagonal fracture network: the laminar flow stage, turbulent flow development stage and turbulent flow stage. In addition, the tracer test reveals that the dominant flow effect is obvious in the hexagonal fracture network.

The experimental results of the flow rate of the hexagonal fracture network under the different pressure drop conditions, the calculation results of the local cubic law and the calculation results of the E-loss model are comprehensively shown in Fig. 10.

Fig. 10
figure 10

Comparison diagram of the test results and theoretical calculation results

Full size image

Figure 10 shows that the E-loss model basically reflects the water flow evolution in the fracture network under laminar flow conditions (below the L–T line). The flow rate at each outlet and the total flow rate under turbulent flow condition can be fitted with the Forchheimer equation, and the coefficient of determination R2 of the fitting results is higher than 0.99. Combined with the fitting results of the quadrilateral fracture network, the quadratic fitting equation suitably describes the seepage characteristics of the fracture network at the turbulent stage.

Analysis of the numerical simulation results

The vector graph directly shows the direction and size of a given vector (such as the velocity) in two-dimensional or three-dimensional space. The velocity vector diagram is an effective means to reflect the velocity change, vortices, reflux conditions, etc. It is one of the most commonly adopted maps in flow field analysis. By default, the vector is drawn at the center of each grid cell, the direction of the vector is indicated with an arrow, and the size of the vector is indicated by the length and color of the arrow. Different colored lines are applied to represent the trajectory of particles and visualize the flow of massless particles in the computational domain. The user may specify the surface from which the particles are released. By describing the streamline diagram and velocity vector diagram of the two models under turbulent conditions, the motion state of the fluid and the distribution of the velocity field in the fluid domain in response to a sudden change in the gap width can be clearly defined. Velocity vector distribution diagrams of the quadrilateral and hexagonal models under the condition of 150 Pa are shown in Figs. 11 and 12, respectively. In both models, three fracture junctions (gap width mutation) are selected, the symmetrical cross sections along the thickness direction are cut to obtain the mesoscale streamline diagram in the plane, and the streamline diagram is displayed on both sides of the vector diagram.

Fig. 11
figure 11

Velocity vector diagram and mesoscale streamline diagram of the quadrilateral fracture network: a frature gap width contraction zone, b frature gap width expansion zone, c frature gap width contraction zone

Full size image
Fig. 12
figure 12

Velocity vector diagram and mesoscale streamline diagram of the hexagonal fracture network: a frature gap width contraction zone, b frature gap width contraction zone, c frature gap width expansion zone

Full size image

In the velocity vector diagram and streamline diagram of the fracture network, the flow direction and velocity distribution of water flow in the two fracture networks are clearly reflected under the same pressure inlet gradient. Under the action of a pressure entrance, an entrance with a larger fracture aperture obtains a higher initial velocity. Choosing the quadrilateral fracture network as an example, the flow velocity at the middle entrance is high, and at the fracture cross junction, due to the contraction effect of the gap width, the flow is diverted to a low-pressure area under the constraint of the wall. This diverted fluid collides with the low-velocity fluid occurring in the horizontal fracture, as shown in areas a and c in Figs. 11 and 12. Based on the streamline diagram, it is found that vortices are generated at the cross fractures, and the velocity vector diagram shows a complex disordered flow behavior, which is the mesoscopic impact of the seepage resistance effect in response to the abrupt change in gap width via contraction.

Compared with the previous research results, the E-loss model proposed in this paper has a wider scope of application. The concept of equivalent seepage resistance is added to the research results of Zhu et al. Therefore, the complex flow diversion problem of fracture network can be transformed into the series parallel problem of resistance, which expands the application scope of this method; In the application of columnar jointed rock mass, if the change of equivalent seepage resistance caused by the change of fracture opening is ignored, the E-loss model will degenerate into the model proposed by Kong et al. In conclusion, the E-loss model is an improvement of the existing model, and can better reflect the seepage process of complex joint network represented by columnar jointed rock mass.

Conclusion

To investigate the characteristics of nonlinear flow in fracture networks, three aspects of theoretical models, physical model tests and numerical simulations are studied in this paper.

  1. 1.

    An E-loss model which can be used to calculate the flow of each fracture in the fracture network is established, the physical meaning of the model is clear and the calculation process is simple.

  2. 2.

    The physical model test results show that the water pressure drop is highly nonlinear with the flow rate in the fracture networks. The fracture connection mode exerts an important influence on the fracture hydraulic characteristics, which leads to a greater inertial energy loss. The test results show that the E-loss model better reflects the flow diversion occurring in the fracture network under a small water pressure drop, and the calculated values of the model are in good agreement with the measured values. With increasing water pressure drop, the Forchheimer equation adequately describes the observed nonlinear flow behavior in the fracture networks.

  3. 3.

    The numerical simulation results reveal that the dominant flow effect is obvious in the two fracture networks. In response to a sudden decrease in fracture width in the fracture network, there is an obvious eddy current blocking effect. This effect is the external impact of the local loss resistance under mesoscale conditions, and the mesoscale water flow state is consistent with the analysis results of the theoretical model.

In seepage analysis of fractured rock mass engineering, attention should be paid to the influences of the fracture connection mode and nonlinear seepage on the fracture hydraulic characteristics. In addition, under the existing theory, because the nonlinear flow effect and local head loss effect are ignored, the water inflow predicted with the linear Darcy equation under a high water pressure at great buried depths may be overestimated.