Influence of particle size on nonDarcy seepage of water and sediment in fractured rock
 Yu Liu^{1, 2}Email author and
 Shuncai Li^{1}
DOI: 10.1186/s4006401637789
© The Author(s) 2016
Received: 1 June 2016
Accepted: 1 December 2016
Published: 20 December 2016
Abstract
Surface water, groundwater and sand can flow into mine goaf through the fractured rock, which often leads to water inrush and quicksand movement. It is important to study the mechanical properties of water and sand in excavations sites under different conditions and the influencing factors of the water and sand seepage system. The viscosity of water–sand mixtures under different particle sizes, different concentration was tested based on the relationship between the shear strain rate and the surface viscosity. Using the selfdesigned seepage circuit, we tested permeability of water and sand in fractured rock. The results showed that (1) effective fluidity is in 10^{−8}–10^{−5} m^{n+2} s^{2−n}/kg, while the nonDarcy coefficient ranges from 10^{5} to 10^{8} m^{−1} with the change of particle size of sand; (2) effective fluidity decreases as the particle size of sand increased; (3) the nonDarcy coefficient ranges from 10^{5} to 10^{8} m^{−1} depending on particle size and showed contrary results. Moreover, the relationship between effective fluidity and the particle size of sand is fitted by the exponential function. The relationship between the nonDarcy coefficient and the particle size of sand is also fitted by the exponential function.
Keywords
Fracture Water–sand NonDarcy seepage NonNewton fluidBackground
From the mechanical perspective, the result of water and sand erupting, permeating fractured rock reflects the instability of the strata layers. Therefore, studying the seepage properties of fractured rocks plays an important role in coal mining engineering. The inrush of water and sand compromises mine safety by causing instability in stress block beams, which creates surface subsidence and water resource run off.
Field tests that are conducted in order to replicate water and sand inrush are difficult; therefore, many scholars suggested conducting experimental simulations of inrushing water and sand. Yang (2009), Yang et al. (2012) and Sui et al. (2007) analyzed the angle of fluid using cemented sand to analyze the mechanisms supporting the inrushing of water and sand. The flow law was examined during various conditions and critical hydraulic gradients of sand inrush currents. Sui et al. (2008) and Xu et al. (2012) analyzed the initial position of inrushing sand based on the structure of water inrush.
Based on underground water dynamic theories, Zhang et al. (2006) created the critical condition and forecasting formula for the prevention of sand inrush by calculating the hydraulic head. Wu (2004) designed a mechanical model of sand inrush pseudo structures, and discussed the force during sand inrush and described the theory of expression of sand inrush. Zhang et al. (2015a) used a case study to discuss drills resulting in sand inrush based on the funnel model. Zhang et al. (2015b) studied the relationship between backfill and water through conducting crack zone. Moreover, river sediment engineering, the theory of sediment transmission and sediment transport mechanics are excellent subject matters to aid in studying the start and movement of sand in mines. Furthermore, the study of sediment engineering, sediment transport theory and practice, and sediment kinematics can aid in understanding the commencement, flow and inrushing sand problem (Du 2014). But others discussed water and sand form the pressure, water and sand flow in tunnel or broken rock (Limin et al. 2016; Du 2014), but the important is seepage in the fracture, which has not been discussed. The concentration and particle’s influence on water and sand inrush.
In this work, permeability attributes of water–sand mixtures are obtained through testing by replicating the design system of water–sand seepage in fractures. The influence of mass concentration in water and particle size of sand on the seepage parameters are tested using specially designed instruments.
Viscosity test of water and sediment
By changing the rotational speed of the NDJ8S viscosimeter, several values of shear strain rate γ and shear stress were obtained and plotted on a \(\gamma  \tau\) scatter diagram. According to the shape of the \(\gamma  \tau\) diagram, the water–sand mixture was identified as nonNewton fluid, and the viscous parameter of water–sand was obtained through linear regression.
Angle strain rate, apparent viscosity and shear stresses at different rotating speed
Rotational speed (rpm)  Shear strain rate (s^{−1})  Apparent viscosity (Pa s)  Shear stress (Pa) 

6  5.24  0.0013  0.0687 
12  10.47  0.0075  0.0785 
30  26.17  0.0034  0.088967 
60  52.33  0.002  0.104667 
Relation of surface viscosity and shearing rate
Particle size (mm)  Concentration (kg/m^{3})  Regression equation  Power exponent  Consistency coefficient (N S^{n}/m) 

0.038–0.044  20  \(\mu_{a} = 0.0569\gamma_{{}}^{  0.7613}\)  0.2387  0.0569 
40  \(\mu_{a} = 0.0617\gamma_{{}}^{  0.8046}\)  0.1954  0.0617  
60  \(\mu_{a} = 0.0623\gamma_{{}}^{  0.8401}\)  0.1599  0.0623  
80  \(\mu_{a} = 0.0701\gamma_{{}}^{  0.8692}\)  0.1308  0.0701  
0.061–0.080  20  \(\mu_{a} = 0.0413\gamma_{{}}^{  0.8235}\)  0.1765  0.0513 
40  \(\mu_{a} = 0.5016\gamma_{{}}^{  0.8433}\)  0.1567  0.0516  
60  \(\mu_{a} = 0.0527\gamma_{{}}^{  0.8744}\)  0.1256  0.0527  
80  \(\mu_{a} = 0.0547\gamma_{{}}^{  0.9091}\)  0.0909  0.0547  
0.090–0.109  20  \(\mu_{a} = 0.0487\gamma_{{}}^{  0.8358}\)  0.1642  0.0487 
40  \(\mu_{a} = 0.0509\gamma_{{}}^{  0.8672}\)  0.1328  0.0509  
60  \(\mu_{a} = 0.0513\gamma_{{}}^{  0.8996}\)  0.1004  0.0513  
80  \(\mu_{a} = 0.0525\gamma_{{}}^{  0.9208}\)  0.0792  0.0525  
0.120–0.180  20  \(\mu_{a} = 0.0416\gamma^{  0.9159}\)  0.0841  0.0416 
40  \(\mu_{a} = 0.0437\gamma_{{}}^{  0.9258}\)  0.0742  0.0437  
60  \(\mu_{a} = 0.0489\gamma_{{}}^{  0.9298}\)  0.0702  0.0489  
80  \(\mu_{a} = 0.0507\gamma_{{}}^{  0.9369}\)  0.0631  0.0507 
Different consistencies were tested of coefficient C and power exponent n with the diameters of sand particle sizes 0.038–0.044, 0.061–0.080, 0.090–0.109 and 0.120–0.180 mm; and sand 20, 40, 60 and 80 kg/m^{3} in the water. The testing results of consistency coefficient C and power exponent n are shown in Table 2.
From Table 2, consistency coefficient C increases with mass concentration in water as exponential relationship, and decreases along with the increase of sand particle; power exponent n increases along with the increase of mass concentration in water, and decreases along with the increase of sand particle.
Seepage test of water and sand in a fracture
Test principle
In the paper, Q is \(6.00 \times 10^{  4} {}3.10 \times 10^{  3}\) m^{3}/s, \(\rho = 1.02{}1.08 \times 10^{3}\) kg/m^{3}, \(\mu = 1.005\;{\text{mp}}_{\text{a}} \;{\text{s}}\).
Introducing the sign \(\lambda_{1} = \frac{1}{{I_{e} }}\left( {\frac{1}{bh}} \right)^{n}\), \(\lambda_{2} = \frac{m\beta }{{(hb)^{2} }}\),
In the test, 5 flows were set as \(Q_{i}^{{}} ,i = 1,2, \ldots ,5\). Steady state values of inlet pressures were tested, and coefficients \(\lambda_{1}^{{}}\) and \(\lambda_{2}^{{}}\) were fitted. The specific process was as follows:
\(\lambda_{1}\) and \(\lambda_{2}\) were solved by Eq. 16, effective mobility \(I_{e}\) and nonDarcy \(\beta\) were obtained.
Experimental equipment and steps
 1.
The test system was assembled according to Fig. 6 and the sample was loaded. The leakage of the experiment system was tested.
 2.
The sand grain with a diameter of 0.038–0.044 mm was placed into the mixing pool and the sand concentration was 20 kg/m^{3} in water.
 3.
To control the motor speed, flow and pressure under different rotational speeds were recorded while the fracture aperture 0.75 mm; the motor speeds, 200, 400, 600, 800, 1000 r/min were changed separately. Different pressures and seepage velocities of the fracture were obtained using a paperless recorder. The sand concentration \(\rho_{s}\) in water was 40, 60, 80 kg/m^{3} respectively.
 4.
The flow and pressure under different grain diameters (0.038–0.044, 0.061–0.080, 0.090–0.109 and 0.120–0.180 mm)were recorded during the different rotational speeds. In order to easily calculate the data, we choose the arithmetic mean of each range of the grain diameter, e.g. 0.041, 0.071, 0.100 and 0.150 mm.
 5.
Results
Pressure graduate
Relationship between pressure gradient and velocity under different sand concentration
Number  Concentration (kg/m^{3})  Pressure gradient (MPa/m)  Velocity (m/s)  Polynomial function  Power function 

1  20  0.38  0.10  \(G_{p} = 155.61V^{2}  \, 41.24V + \, 3.72\) R^{2} = 0.9882  \(G_{p} = 0.29e^{9.04V}\) R^{2} = 0.9259 
2.10  0.17  
5.08  0.27  
6.98  0.35  
15.27  0.44  
24.92  0.52  
4.54  0.17  
2  40  7.62  0.30  \(G_{p} = 118.08V^{2}  \, 36.42V \, + \, 7.43\) R^{2} = 0.9970  \(G_{p} = 2.14e^{4.28V}\) R^{2} = 0.9954 
11.11  0.40  
19.21  0.50  
27.98  0.60  
5.71  0.11  
3  60  9.30  0.26  \(G_{p} = 108.12V^{2}  \, 7.42V \, + 5.38\) R^{2} = 0.9956  \(G_{p} = 3.74e^{4.07V}\) R^{2} = 0.9953 
15.24  0.35  
24.16  0.40  
33.81  0.51  
7.94  0.16  
12.70  0.27  
4  80  22.22  0.35  \(G_{p} = 55.04V^{2} + 119.60V  11.08\) R^{2} = 0.9602  \(G_{p} = 4.50e^{4.35V}\) R^{2} = 0.8968 
28.57  0.44  
35.24  0.51 
Figure 7 and Table 3 been presented above, we can obtain: the seepage velocity of water and sand increases with pressure gradient increasing, Moreover, the greater the sand concentration in water is, the lower the seepage velocity is.
Permeability of water and sand in the fracture
Because of the permeability parameters of water and sand seepage in fracture are connected with water and sand, at the same time, the structure of fracture; so the permeability k is not enough to describe permeability parameters, the effective fluidity and nonDarcy factor \(\beta\) are used.
permeability parameters of water and sand under different sand concentration
Concentration of sediment (kg/m^{3})  Particles size  \(I_{e}\) \(I_{e} \;\left( {{\text{m}}_{{}}^{n + 2} \cdot {\text{s}}_{{}}^{2  n} /{\text{kg}}} \right)\)  \(\beta \;({\text{m}}_{{}}^{  1} )\) 

20  0.04  1.57E−06  5.67E+06 
0.075  7.62E−07  1.46E+07  
0.100  4.34E−07  2.64E+07  
0.151  1.71E−07  6.48E+07  
40  0.04  8.34E−07  2.14E+06 
0.075  5.53E−07  1.03E+07  
0.100  3.54E−07  1.75E+07  
0.151  1.73E−07  2.93E+07  
60  0.04  6.82E−07  3.04E+06 
0.075  4.41E−07  1.10E+07  
0.100  2.60E−07  2.54E+07  
0.151  6.31E−08  9.33E+07  
80  0.04  6.76E−07  1.05E+06 
0.075  6.76E−07  4.77E+07  
0.100  2.20E−07  9.90E+07  
0.151  1.28E−07  2.28E+08 
Fitted equations of permeability parameters changing with \(d_{s}\) at JRC 4–6
Number  Sand concentration (kg/m^{3})  Permeability parameters  Fitting equations  Coefficient 

1  20  \(I_{e}\)  \(I_{e} = 3.40 \times 10^{  6} e^{{  20.03d_{s} }}\)  0.9980 
\(\beta\)  \(\beta = 2.64 \times 10^{6} e^{{21.79d_{s} }}\)  0.9872  
2  40  \(I_{e}\)  \(I_{e} = 1.65 \times 10^{  6} e^{{  14.80d_{s} }}\)  0.9693 
\(\beta\)  \(\beta = 3.10 \times 10^{5} e^{{33.82d_{s} }}\)  0.7805  
3  60  \(I_{e}\)  \(I_{e} = 1.95 \times 10^{  6} e^{{  21.81d_{s} }}\)  0.9688 
\(\beta\)  \(\beta = 1.01 \times 10^{6} e^{{30.66d_{s} }}\)  0.9905  
4  80  \(I_{e}\)  \(I_{e} = 1.15 \times 10^{  6} e^{{  15.10d_{s} }}\)  0.9781 
\(\beta\)  \(\beta = 1.78 \times 10^{5} e^{{48.84d_{s} }}\)  0.9549 
The exponential function was used to fit the relationship between effective fluidity, nonDarcy coefficient and particle sizes of sand. The power exponent equations are used to fit the relationship between effective fluidity I _{ e }, the nonDarcy factor \(\beta\) and Sand concentration.
 1.
The seepage of water and sand in a fracture is nonlinear.
 2.
Along with the change of grain size of sediment, the relationship between effective fluidity \(I_{e}\) and mass concentration of sand \(d_{s}\) was the negative exponential relationship; the absolute value of the exponent increased along with the increase of sand particle in the water.
 3.
NonDarcy factor β and sand concentration in water had a positive exponential relationship; the absolute value of the exponent increased along with the decrease of sand particle in water.
Discussion
It is nonDarcy flow in the paper, which was influenced by roughness, flow velocity, aperture of fracture, and so on. Roughness has a large influence on fracture flow, where nonDarcy also happened (Boutt et al. 2006; Lomize 1951; Louis 1969; Qian et al. 2011).
During the flow, Reynolds number and Forchheimer’s number are important parameters to judge (Bear 1972): when Re > 100 or Re < 1, it will be nonlinear flow and does not conform to Darcy flow. What’s more, the velocity of water and sand, the aperture of fracture and the tortuosity of fracture also have much influence on flow parameters (Tsang 1984; Tsang and Tsang 1987). The concentration and density also have influence on flow character in fracture (Watson et al. 2002; Tenchine and Gouze 2005). Here \(I_{e}\) has relationship with the structure of fracture, and the character of mixture or water and sand. With the pressure drop increasing, the nonlinear flow became obvious (Elsworth and Doe 1986; Wen et al. 2006; Yeo and Ge 2001) the Forchheimer’s law is well known classical approach to describe the nonlinear flow in fracture. NonDarcy factor β is the parameter which reflected the deviation of Darcy of the seepage. Along with sand particle in water, the nonDarcy character became more obvious.
Conclusion
 1.
The seepage velocity of water and sand in a fracture increases along with the pressure of the fracture, but the relationship between them is nonlinear.
 2.
Consistency coefficient C becomes larger in conjunction with the mass concentration in water, but decreases along with the particle size of sand. The lower exponent n becomes enlarger along with mass concentration in water, but decreases along with particle size of sand.
 3.
Along with the change of the grain size of sediment, the relationship between effective fluidity \(I_{e}\) and mass concentration of sediment \(\rho_{s}\) in water is exponential. The absolute value of the exponent increases along with the increase of sand concentration in water. The nonDarcy factor β and sand concentration in water has a positive exponential relationship and the absolute value of the exponent increases along with the decrease of sand concentration in water.
 4.
For the future work, we will work for the different concentration, for particle and concentration both has influence to the flow character, but we should do some experiments to make sure which one is more influence. And acceleration, low velocity of water and sand how to change into water and sand inrush.
List of symbols
 b :

aperture of fracture
 C :

consistency coefficient
 C _{ a } :

acceleration coefficient
 d :

diameter of the rotor
 D :

diameter of outer cylinder
 d _{ s } :

particle size of sand
 h :

height of the fracture
 I _{ e } :

effective fluidity
 k _{ e } :

effective permeability
 L :

sample length
 m :

mass of sand and water
 n :

power exponent
 n _{ rot } :

rotate speed of rotor
 β :

nonDarcy coefficient
 p :

pressure
 Q :

flow of seepage
 \(\tau\) :

shear stress
 \(\mu\) :

fluid viscosity
 \(\mu_{a}\) :

apparent viscosity of water and sand
 \(\mu_{e}\) :

effective viscosity
 V :

velocity of seepage
 \(\gamma\) :

apparent viscosity
 \(\rho\) :

density
 \(\rho_{s}\) :

mass concentration of sand
 \(\frac{\partial p}{\partial l}\) :

pressure gradient
Declarations
Authors’ contributions
The work presented here was carried out in collaboration between all authors. Yu Liu defined the research theme, designed experiments methods and wrote the paper. Shuncai Li did the experiments, analyzed the data and explained the results. Both authors read and approved the final manuscript.
Acknowledgements
This research was supported by Natural science fund for colleges and universities in Jiangsu Province (14KJB440001), Jiangsu Normal University PhD Start Fund (14XLR032), Jiangsu Planned Projects for Postdoctoral Research Funds (1402055B), and National Natural Science Foundation of China (51574228), All the supports are gratefully acknowledged.
Competing interests
Both authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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