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Rock fractures always influence the hydrological properties of a rock mass. To investigate the seepage characteristics of a rock mass with partly filled fractures, a mathematical model is established. In this model, the clear fluid in fractures is governed by the Navier-Stokes equation, and the fluid both in the porous medium and rock matrix are subjected to the Brinkman-Extended Darcy equation. The analytic solution of an equivalent permeability coefficient for a rock mass with partly filled fractures is solved, and it could be reduced to some special known results. Comparisons with experimental data show good agreement, thus verifying the validity of the present computations.

Natural rock masses are composed of intact rock blocks with numerous fractures. Rock fractures can greatly influence the hydrological properties and control the mechanical behaviour of a rock mass. The hydraulic and mechanical properties of rock fractures are of considerable interest in several areas of rock engineering, such as near-field modelling of hydraulic processes around radioactive waste repositories [

At present, detailed studies on the permeability of a rock mass with an unfilled fracture have been conducted, and they focus on the influences of stress [

Actually, natural fractures always contain infilling material, which affects the overall permeability of the fractured rock mass. Takemura et al. [

In a natural geological body, there are not only completely filled fractures and empty fractures but also partly filled fractures. However, there are few studies that have investigated seepage in a rock mass with partly filled fractures, despite much research on the seepage of fractured rock masses having been conducted. In this paper, a nonlinear mathematical model is proposed. In this model, the motion of fluid in a clear fracture is governed by the Navier-Stokes equation, and seepage flow in infilling material and rock matrix is subjected to the Brinkman-Extended Darcy equation. Through analysis of the results, the flow velocities of fluid in open and filled fractures and the rock matrix were derived, as well as an analytical solution for the equivalent permeability coefficient (

In reality, a rock mass is not intact, as substantial fissures or fractures are present that were generated over a long period of geological history. Some fractures are filled with debris and sediment, some are open, and others are partly filled. Clearly, the permeability of a rock mass varies with the degree of fracture fill, which may also exert a significant impact on the safety of underground tunnels and the stability of rock slopes. To get a better understanding of the permeability characteristics of a rock mass with partly filled fractures, a periodical seepage analytical model was designed using a Cartesian coordinate system (

Fluid flow is infinite along the

The fluid is a fully developed Newtonian fluid

The fluid is incompressible, and the volume force along the

The fluid is viscous with laminar flow

The fracture/crack extends infinitely in the

Velocity and shear stress are continuous at the interface between different media

The fluid in fractures, fillings, and the rock matrix satisfies the continuity equation

Seepage through crack fillings and the rock matrix satisfies the Brinkman-Extended Darcy equation

Model of rock mass with periodic partially filled fractures.

The clear fluid in fractures can be described by the Navier-Stokes equation:

The description of the interface conditions between a clear fluid and a porous medium has received considerable attention. In summary, three primary categories of conditions at the interface were found in the literature; these can be classified into two main conditions: slip and no slip [

Beavers and Joseph [

Neale and Nader [

Ochoa-Tapia and Whitaker [

Furthermore, we find that the interface description of each category is not a closed system of equations. The success of each model is dependent on the accuracy of a free integration constant, i.e.,

Based on the theoretical model and associated assumptions, a period unit was selected for analysis, as shown in Figure

The clear fluid in the fracture is governed by the continuity equation:

The motion equation satisfies the Navier-Stokes equation:

As the fluid velocities are 0 in the

According to the condition that the gradation variation of

Together with the two conditions of

By substituting

The solution for equation (

The fluid in fillings satisfies the continuity equation:

The motion equation satisfies the Brinkman-Extended Darcy equation:

Similar to the above analysis for equation (

By solving equation (

Fluid in the rock matrix satisfies the continuity equation:

The motion equation satisfies the Brinkman-Extended Darcy equation:

Similar to the above analysis for equation (

By solving equation (

According to the mathematical model, the following boundary conditions are satisfied:

At the interface between different media,

At the interface between different media,

At the interface between different media,

These boundary conditions could be expressed as follows:

Substitute these boundary conditions into equations (

Parameters

The seepage field in a layered rock mass is a three-dimensional anisotropic field. In the horizontal plane, the flow velocities are the same. The permeability in the horizontal plane can be solved using the theory described above.

As shown in Figure

Darcy’s law states that

Additionally, we have

The equivalent permeability of a rock mass can be obtained by solving equations (

The analytic solution for the equivalent permeability coefficient (

As shown in Figure

Fractured rock seepage model along the

As the flow velocity in each layer is equal, it can be described as follows:

And because the total water head loss should be equal to the sum of the water head loss in each layer, we have

According to equations (

As

And the permeability of a periodically partially filled rock mass along the

Equation (

To verify the above analysis, a penetration device was designed; it is shown in Figure

Seepage testing equipment for the fractured rock mass.

In this study, a porous stone produced by Hongzhou Experimental Instrument Co. Ltd. was used as a substitute for the rock matrix. For the sampling procedure, waterproof glue was applied to the inner surface of the organic box to avoid permeation, and the porous stone was fixed by applying waterproof glue (

The main steps during the test procedure are as follows:

Turning on the switch of the water inlet device and water will flow into the seepage device and slowly get the testing sample saturated

Clamping the discharging tube by a water stop clamp and adjusting the altitude of the water inlet device to get the designated water head difference, while the seepage device was full of water

Removing the water stop clamp and beginning to record the time for the specified amount of seepage after the flow became steady

Constant head permeability tests were carried out as per GB/T 50123-1999 [

Testing device used in the constant head permeability test.

The permeability coefficient can be determined using
^{2}), and

The dry density of sand (^{3}. The constant head permeability test results are shown in Table

Results of the penetration test for sand.

Number | Time |
Water head difference of piezometer tube (cm) | Water head difference (cm) | Hydraulic gradient |
Volume ^{3}) |
Permeability coefficient |
Average value | ||||
---|---|---|---|---|---|---|---|---|---|---|---|

I | II | III | Average value | ||||||||

— | (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) |

— | — | — | — | — | (2)-(3) | (3)-(4) | — | — | |||

1 | 25 | 251 | 178 | 101 | 73 | 77 | 75 | 7.5 | 215 | ||

2 | 25 | 228 | 145 | 50 | 83 | 95 | 89 | 8.9 | 220 | ||

3 | 25 | 218 | 133 | 36 | 85 | 97 | 91 | 9.1 | 245 | ||

4 | 25 | 205 | 117 | 20 | 88 | 97 | 92.5 | 9.25 | 256 |

According to

The porosity of the sand fill can be obtained as follows [^{3}),

By substituting these parameters into equation (

The experiment was carried out using the procedure presented above. In the experiment, three kinds of foam were used, with thicknesses of 1 mm, 2 mm, and 3 mm. The results for these tests are shown in Tables

Results of the penetration test for a fractured rock mass (

Number | Seepage path |
Area of section ^{2}) |
Volume ^{3}) |
Water head difference |
Time |
Permeability coefficient |
Average value |
---|---|---|---|---|---|---|---|

1 | 20 | 400 | 6000 | 15 | 33 | 0.606 | 0.592 |

6000 | 20 | 25 | 0.6 | ||||

6000 | 25 | 21 | 0.571 | ||||

2 | 20 | 400 | 7000 | 15 | 39 | 0.598 | 0.582 |

7000 | 20 | 31 | 0.564 | ||||

7000 | 25 | 24 | 0.583 | ||||

3 | 20 | 400 | 8000 | 15 | 45 | 0.593 | 0.584 |

8000 | 20 | 34 | 0.588 | ||||

8000 | 25 | 28 | 0.571 |

Results of the penetration test for a fractured rock mass (

Number | Seepage path |
Area of section ^{2}) |
Volume ^{3}) |
Water head difference |
Time |
Permeability coefficient |
Average value |
---|---|---|---|---|---|---|---|

1 | 20 | 400 | 50000 | 15 | 47 | 3.58 | 3.61 |

50000 | 15 | 45.5 | 3.70 | ||||

50000 | 15 | 49 | 3.55 |

Results of the penetration test for a fractured rock mass (

Number | Seepage path |
Area of section ^{2}) |
Volume ^{3}) |
Water head difference |
Time |
Permeability coefficient |
Average value |
---|---|---|---|---|---|---|---|

1 | 20 | 400 | 100000 | 15 | 28 | 11.90 | 11.84 |

100000 | 15 | 29 | 11.49 | ||||

100000 | 15 | 27.5 | 12.12 |

By substituting the parameters of the sand fill (

Comparison between the experimental and computed results.

Number | Error (%) | |||||
---|---|---|---|---|---|---|

1 | 20 | 1 | 0.9 | 0.54 | 0.586 | 7.8 |

2 | 20 | 1 | 0.8 | 3.32 | 3.61 | 8.0 |

3 | 20 | 1 | 0.7 | 10.96 | 11.84 | 7.4 |

If

This result agrees with a well-known result reported earlier by Arbogast and Lehr [

If

If

This result is in good agreement with the well-known result reported by Lomize [

Here, we use the Yuelongmen tunnel of the Chengdu-Lanzhou railway as an applied example of this method. This tunnel is located in the Longmenshan fault zone. The total length of the tunnel is 19.981 km. The main rock types at the tunnel site are limestone, dolomite, and mudstone with occasional granulite, sandstone, and quartzite.

As a result of crossing the Longmenshan fault zone, especially after undergoing the Wenchuan earthquake with Ms 8.0 on May 12th, 2008, the rock mass became relatively fractured, and joints and fissures developed widely. According to a field investigation, the seepage volume at a location 9.4 km from the entrance of the Yuelongmen tunnel amounted to 1682 m^{3}/d, and problems such as serious water gushing, water in-rush, and mud outbursts occasionally occurred when construction was occurring on these fault sections and fissure development zones, which presented a serious obstacle to construction.

To investigate seepage while drilling a tunnel in a zone of fissured rock, a simulation device was designed, as shown in Figure ^{3} kg/m^{3}.

Physical model and its local details.

Physical model and key sizes.

Local position photos of the tunnel and the plug in the physical model.

Seepage volume at different times.

Time (s) | 60 | 120 | 180 | 300 | 480 | 690 | 900 | 1200 | 1500 | 1800 | 2400 | 3000 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Seepage volume (10^{3} cm^{3}) |
0.17 | 0.68 | 1.01 | 1.59 | 3.04 | 4.12 | 5.38 | 7.05 | 8.15 | 9.97 | 15.12 | 21.27 |

The results from the test and computation based on this method are shown in Figure

Comparison between the calculated values and test results.

Rock fractures can greatly influence the hydrological properties of and control the mechanical behaviour of the rock mass. To investigate the seepage characteristics of a rock mass with partly filled fractures, a modified model from the Cubic Law is established, which assumes that a clear fluid in a fracture is governed by the Navier-Stokes equation and that the fluid both inside the porous medium and the rock matrix are subject to the Brinkman-Extended Darcy equation. Through analysis, the analytic solution for the equivalent permeability coefficient (

The results show that the equivalent permeability coefficient (

In conclusion, the analytical solution of the equivalent permeability coefficient for a rock mass with partially filled fractures is more comprehensive and reasonable, and it can not only be adopted for calculating the permeability coefficient of a rock mass with filled fractures and unfilled fractures but can also be used for partially filled fractures.

The flow velocities in infilling sand and in the rock matrix are

According to the analysis in Section

The following three boundary conditions are given:

At

At

Solving equations (

At

Solving equations (

Combining equations (

As

Thus, the average flow velocity (

Substituting equations (

As

Substituting equation (

Substituting equation (

The data used to support the findings of this study are included within the article.

The authors declare that there is no conflict of interest regarding the publication of this paper.

The contributions of Prof. Fu included conceptualization; funding acquisition; project administration; supervision; and writing, reviewing, and editing. The contributions of Dr. Ye included conceptualization, data curation, investigation, methodology, validation, and writing the original draft. The contributions of Dr. Duan included conceptualization, data curation, and investigation. The contributions of Dr. Yuan included data curation and investigation. Dr. Duan was our teammate and has graduated from Sichuan University last June 2018. However, when we submitted the original manuscript in August, 2018, we forgot to add his name.

The authors thank Elsevier’s Webshop for editing and polishing this paper. This work was supported by the International Cooperation Project of Sichuan Province (Grant No. 2018HH0082); the National Nature Science Foundation of China (Grant No. 41772321); the Science and Technology Planning Project of Sichuan Province, China (No. 2017TD0007); and the National Key R&D Program of China (Grant No. 2018YFC1505004).