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The mechanism and quantitative descriptions of nonlinear fluid flow through rock fractures are difficult issues of high concern in underground engineering fields. In order to study the effects of fracture geometry and loading conditions on nonlinear flow properties and normalized transmissivity through fracture networks, stress-dependent fluid flow tests were conducted on real rock fracture networks with different number of intersections (1, 4, 7, and 12) and subjected to various applied boundary loads (7, 14, 21, 28, and 35 kN). For all cases, the inlet hydraulic pressures ranged from 0 to 0.6 MPa. The test results show that Forchheimer’s law provides an excellent description of the nonlinear fluid flow in fracture networks. The linear coefficient

Rock fracture networks constitute the main pathways of fluid flow and solute migration in deep underground projects, and during the past several decades, substantial efforts have been devoted to the estimation of fluid flow behavior and transmissivity of fractures in many geoengineering and geosciences such as underground tunneling [_{2} sequestration [

The transmissivity

Relationships between hydraulic gradient

In engineering practices, both nature activities and human perturbations entail significant changes in the effective stress field of underground rock masses, which would then impact the fracture aperture in fracture networks. The void space between opposing surfaces can vary due to normal stress-induced closures or opening [

The purpose of this paper is to investigate the stress-dependent nonlinear flow properties and normalized transmissivity of real rock fracture networks. First, plate granite specimens containing fracture networks with different number of intersections (1, 4, 7, and 12) were machined using a high-pressure water jet cutting system. Next, a series of high precision stress-dependent water flow tests with respect to different inlet hydraulic pressures ranging from 0 to 0.6 MPa and increasing applied boundary loads from 7 to 35 kN were conducted. The nonlinear flow behaviors, variations of the flow nonlinearity, normalized transmissivity and equivalent permeability of the fracture networks as a function of fracture networks geometry, and boundary load conditions were all examined.

Fluid flow through a single fracture is generally governed by the following Navier-Stokes (NS) equations, written in a tensor form as [

For the case of incompressible and steady-state Newtonian flow, the terms involving time

Equation (

In (

For certain cases with very low Reynolds number or flow rate, by assuming that the inertial forces of fluid flow through the fractures are negligible compared to the viscous forces. The convective acceleration term

The linear relation between

Some empirical expressions were proposed to describe the nonlinear flow in fractures, and Forchheimer’s law is the most extensively used approach where the pressure gradient is a quadratic function of the flow rate, written as [

Since the hydraulic gradient

The hydraulic gradient

To quantitatively estimate the nonlinearity of fluids flowing through the fracture networks, a nonlinear effect factor

For engineering purposes, a critical value of

The Forchheimer number

Combinations of (

Transmissivity (

When the flow rate is sufficiently low and the inertial forces are negligible, the intrinsic transmissivity (

Therefore, as the nonlinear term (

Granite specimens (495 × 495 × 17 mm in size) containing artificial fracture networks were established by using a high-pressure water jet cutting system [^{3}. The uniaxial compressive strength of the rock is about 97.54 MPa, and its permeability is in the order of magnitude of ^{2}.

The parallelism between the upper and lower surfaces of the plate specimens was controlled within an error of 0.02 mm. The water pressure of the water jet cutting system was held constant for all rock fractures, and the fractures penetrated the plate specimens thoroughly. Detailed descriptions of the rock fracture specimens are illustrated in Figure

Plate granite specimens of fracture networks with different numbers of intersections (

Using a high-resolution three-dimensional laser scanning profilometer system, the fracture surface topography was measured before the hydromechanical tests. We considered the average of joint roughness coefficient (JRC) values for a series of 2D profiles along a fracture surface in the length direction, which is a suggested method by the International Society for Rock Mechanics and Rock Engineering (ISRM) to calculate the JRC of a 3D single fracture [

The results indicate that JRC values of the fractures fluctuate within a very small range around 3.47. The fractures closely approximate parallel-plate models with a small JRC value.

Before the hydromechanical tests, sealing of the fractured specimens was first conducted. Based on the model specimen size, a 3 mm thick rubber jacket was produced using the ethylene propylene diene monomer (EPDM) waterproof rubber [

The stress-dependent flow tests of the fracture networks were conducted by using a self-developed flow test apparatus [

Schematic view of the stress-dependent fluid flow test apparatus.

(a) A top view of hydraulic setup of the test apparatus. (b) Loading diagram of the rock fracture specimen.

During the flow test, water was fed through the inflow manifold connected to a water tank that can supply a constant water head at all times by using an air compressor. Both horizontal boundary loads in the

For a certain boundary load and inlet hydraulic pressure, the total volume flow rate of the fracture networks at the water outlet boundary can be obtained when the glass rotor was relatively stable with no fluctuations. The effluent was then collected in another storage tank and recirculated. The entire hydraulic experiments were performed under isothermal conditions at room temperature of approximately 20°C. In addition, the fluid was assumed to be viscous with ^{−3 }Pa·s and incompressible with ^{3}.

Due to the low permeability of the intact granite matrix, fluid was assumed to flow through the fractures only during the hydromechanical tests. For the rock fracture specimen with various

The experimental data obtained in the hydraulic tests on the rock fracture specimens in the form of hydraulic gradient (^{−6 }m^{3}/s (^{−6 }m^{3}/s (

Values of

| | ^{−1}⋅s^{−1}⋅m^{−4}) | ^{−1}⋅m^{−7}) | | |
---|---|---|---|---|---|

1 | 7 | | | 0.9993 | 6.01 |

14 | | | 0.9994 | 10.78 | |

21 | | | 0.9993 | 25.34 | |

28 | | | 0.9994 | 57.05 | |

35 | | | 0.9996 | 118.57 | |

4 | 7 | | | 0.9993 | 2.29 |

14 | | | 0.9990 | 4.80 | |

21 | | | 0.9995 | 15.16 | |

28 | | | 0.9995 | 27.17 | |

35 | | | 0.9994 | 59.06 | |

7 | 7 | | | 0.9991 | 1.63 |

14 | | | 0.9995 | 1.96 | |

21 | | | 0.9994 | 9.58 | |

28 | | | 0.9996 | 18.44 | |

35 | | | 0.9975 | 21.50 | |

12 | 7 | | | 0.9994 | 0.62 |

14 | | | 0.9989 | 1.45 | |

21 | | | 0.9995 | 6.96 | |

28 | | | 0.9994 | 8.36 | |

35 | | | 0.9996 | 13.01 |

Regression analysis of hydraulic gradient as a function of measured flow rate using Forchheimer’s law of fracture networks subjected to increased boundary loads.

The regression linear and nonlinear coefficients

Variations in linear and nonlinear coefficients

In view of the fact that the variations of linear and nonlinear coefficients

Although (

To quantify the degree of the nonlinear effect of fluid flow through the fracture networks, variations of

Evolution of nonlinear effect factor

By defining a critical value of

Relationships between

For fluid flow through fractured and porous media, the transmissivity has also been applied to estimate the nonlinear flow regime. Using (^{4} and

Regression analysis of transmissivity (

When assessing the fluid flow through a single rock fracture, the Reynolds number (Re), which is defined as the ratio of inertial forces to viscous forces, is typically used to quantify the onset of nonlinear flow. As noted above, the variations of normalized transmissivity (

However, in engineering practices, there are hundreds to thousands of fractures, and the Re of each fracture generally cannot be ascertained. Instead,

By fitting the experimental data sets, the relationships between

Relationships between hydraulic gradient

From Figure

The variations of the coefficient

Variations in the coefficient

From the quadratic relationships between

Calculations were made with the following equation, assuming no gravity term:

The changes in the equivalent permeability of the fracture networks in terms of

Equivalent permeability

This paper experimentally examines the impacts of number of intersections (

Forchheimer’s law offers a good description for the relationships between volume flow rate and hydraulic gradient of water flow through the rock fracture networks. Both the linear coefficient

The critical hydraulic gradient of the fracture networks is calculated by taking a critical nonlinear effect factor

With an increase in the hydraulic gradient, the transmissivity of the fracture networks displays an exponential decrease trend. In addition, the transmissivity increases with the number of intersections but decreases with the applied boundary load. The variations in normalized transmissivity as a function of hydraulic gradient were estimated with a mathematical expression

The authors declare that they have no conflicts of interest.

The financial support from the Fundamental Research Funds for the Central Universities, China (no. 2018QNA32), is gratefully acknowledged.

_{2}permeability analysis of caprock containing a single fracture subject to coupled thermal-hydromechanical effects