Test principle
Figure 4 demonstrates a model of seepage in a fracture. In this paper, the aperture of fracture is 0.75 mm, the length is 12.5 mm, the height is 75 mm. b is the aperture of fracture, h is the height of the fracture, and L is the sample length.
According to Fig. 4, we can get Eq. 5.
where V is the velocity of seepage, Q is the flow of seepage.
For the fracture, Re is defined as Eq. 6 (Javadi et al. 2014).
$$R_{e} = \frac{\rho Q}{\mu b}$$
(6)
where Re is Reynolds number, ρ is the density, Q is the flow of seepage, μ is the fluid viscosity.
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}, \(\mu = 1.005\;{\text{mp}}_{\text{a}} \;{\text{s}}\).
So, \(R_e = \frac{\rho Q}{\mu b} = 76.5{}421.2\) in case of higher Reynolds numbers (\(R_e \gg 1\)), the pressure losses pass from a weak inertial to a strong inertial regime, described by the Forchheimer equation (Forchheimer 1901; Chin et al. 2009; Cherubini et al. 2012, 2013; Javadi et al. 2010; Li et al. 2008), given by:
$$\rho c_{a} \frac{\partial V}{\partial t} =  \frac{\partial p}{\partial l}  \frac{\mu }{k}V  \rho \beta V^{2}$$
(7)
where \(\mu\) is fluid viscosity, \(\beta\) is nonDarcy factor, the pressure is \(p\), \(\frac{\partial p}{\partial l}\) is the pressure gradient, \(c_{a}\) is the acceleration of water and sand, \(b_{1}\) is two term coefficient.
Because of water and sand permeability parameter’s particularity (permeability parameter is relevant to liquid and fracture), we use \(\mu_{e}\), \(k_{e}^{{}}\) to describe the water and sand of effective viscosity \(\mu_{e}\), effective permeability \(k_{e}^{{}}\), as shown in Eq. 8 (Liu 2014).
$$\rho c_{a}^{{}} \frac{\partial V}{\partial t} =  \frac{\partial p}{\partial l}  \frac{{\mu_{e}^{{}} }}{{k_{e}^{{}} }}V_{{}}^{n}  \rho \beta V_{{}}^{2}$$
(8)
As for one kind of nonNewton fluid, liquid viscosity and permeability in fracture of water–sand mixture were related to fluid properties and fracture aperture. Therefore, liquid viscosity and permeability were not obtained separately, and the effective fluidity \(I_{e}^{{}}\) was introduced to simplify the expression.
$$I_{e}^{{}} = \frac{{k_{e} }}{{\mu_{e} }}$$
(9)
The Eq. 8 can be changed into
$$\rho c_{a}^{{}} \frac{\partial V}{\partial t} =  \frac{\partial p}{\partial l}  \frac{1}{{I_{e}^{{}} }}V_{{}}^{n}  \beta \rho V_{{}}^{2}$$
(10)
Equation 10 calculated the momentum conservation of water–sand seepage in the fracture. For the seepage in Fig. 4, the steadyflow method was selected to measure water–sand seepage in the fracture. Equation 10 can be deduced into Eq. 11,
$$\frac{1}{{I_{e} }}V^{n} + \beta \rho V^{2} =  \frac{\partial p}{\partial l}$$
(11)
Substituting Eq. 5 into Eq. 11 yields Eq. 12
$$ dp = \frac{1}{{I_{e} }}\left( {\frac{Q}{bh}} \right)^{n} dl + \beta \rho \left( {\frac{Q}{bh}} \right)^{2} dl$$
(12)
b is the aperture of the fracture, m is the mass of sand and water.
For the length, the integrating range is [0, L]; the mass is m, the pressure of water and sand at the entrance wall were:
$$\left\{ {\begin{array}{l} {\left. p \right_{x = 0}^{{}} = p_{0}^{{}} } \\ {\left. p \right_{x = L}^{{}} = 0} \\ \end{array} } \right.$$
(13)
The definite integral of Eq. 12 on the interval [0, L] was
$$p = \frac{L}{{I_{e} }}\left( {\frac{Q}{bh}} \right)^{n} + \beta mL\left( {\frac{Q}{bh}} \right)^{2}$$
(14)
Introducing the sign \(\lambda_{1} = \frac{1}{{I_{e} }}\left( {\frac{1}{bh}} \right)^{n}\), \(\lambda_{2} = \frac{m\beta }{{(hb)^{2} }}\),
Therefore, Eq. 14 was obtained by using
$$\lambda_{1} Q^{n} + \lambda_{2} Q^{2}  p_{0} = 0$$
(15)
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:
Equation 15 was obtained
$$\varPi = \sum\limits_{i = 1}^{5} {\left( {\lambda_{1}^{{}} Q_{i}^{n} + \lambda_{2}^{{}} Q_{i}^{2}  p_{0}^{i} } \right)_{{}}^{2} } = 0$$
(16)
In order to get the least value of the flow Q, Eq. 16 can be set as Eq. 17.
$$\left\{ \begin{aligned} \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{n} Q_{i}^{n} } } \right)\lambda_{1}^{{}} + \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} Q_{i}^{n} } } \right)\lambda_{2}^{{}} = \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{n} p_{0}^{i} } } \right) \hfill \\ \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} Q_{i}^{n} } } \right)\lambda_{1}^{{}} + \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} Q_{i}^{2} } } \right)\lambda_{2}^{{}} = \left( {\sum\limits_{i = 1}^{5} {Q_{i}^{2} p_{0}^{i} } } \right) \hfill \\ \end{aligned} \right.$$
(17)
\(\lambda_{1}\) and \(\lambda_{2}\) were solved by Eq. 16, effective mobility \(I_{e}\) and nonDarcy \(\beta\) were obtained.
Experimental equipment and steps
Based on testing principles, a set of experimental system was designed and manufactured as shown in Fig. 5. Sand comes from the surface of the mine in northwest of China. The rock sample is the sandstone under −265 m from the Luan mine in Shanxi, China. There are five specimens of rock fracture with Joint Roughness Coefficient (JRC) 4–6, the velocity of seepage was obtained.
Figure 6 illustrates the entire experimental procedure. The test steps were as follows:

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.
According to Eqs. 15 and 16, \(I_{e}\) and \(\beta\) were calculated.