In the permeability test, the effect of wall effects on the flow behavior of specimens must be considered. In general, the larger the ratio of the diameter of the packed column to the particle size, the smaller the wall effect. Hansen (1992) reported that wall effects could be excluded when the ratio of the diameter of the packed column to the average particle size of grains is larger than 10. However, most permeability testing apparatuses are not suitable for exploring the permeability of phyllite bimrocks because the size of phyllite blocks is large. Therefore, a self-developed large-scale permeability apparatus was designed to test the permeability property of phyllite bimrocks (Figure 3). According to the Code for Coarse-Grained Soil Tests for Hydropower and Water Conservancy Engineering (DL/T5356-2006DL/T5356-2006), the sample height should not be less than the sample diameter. Thus, the diameter (d) and the height (L) of the sample were selected as 40 and 40 cm, respectively.

The experimental apparatus for the permeability test of phyllite bimrocks consists of the water supply part (Part I), water head measurement part (Part II) and discharge collection part (Part III). The overall setup is shown in Figure 3.

The water supply part comprises an inflow and overflow pipe, water tank and height adjustment device. The water flows into the water tank through the inflow pipe. Once the water level reaches a certain height, it flows out through the overflow pipe. By adjusting the height of the water tank, the water can inject at various rates or pressure into the sample. Therefore, different seepage velocities and hydraulic gradients can be set up.

The water head measurement part consists of an acrylic cylinder, cobblestone, acrylic perforated plate, discharge tube, calibration board, inlet valve, and piezometric tube. The water enters the water head measurement system from the bottom of the acrylic cylinder. The cobblestones are filled in the bottom of the cylinder, allowing water to spread from the entry point. The acrylic perforated plate can hold the sample and enable the water to flow into the sample. After the water flow through the sample, the water head difference ( Δ h ) is produced and can be calculated by reading the piezometric tubes’ difference value correspondingly.

The discharge collection part is made up of a measuring cylinder. The seepage water flows out through the discharge tube and is collected by the measuring cylinder. Accordingly, the seepage water volume ( Q ) can be obtained.

In the experimental test, the threshold grain size of phyllite soil and phyllite rocks in phyllite bimrocks must be confirmed. Xu and Hu (2006) called it the soil/rock threshold. Additionally, according to the research by Medley (1994) and Xu and Hu (2006), the soil/rock threshold could be defined as: d S / R T = 0.05 L c (1) where d S / R T is the soil/rock threshold and L c is the characteristic engineering dimension and could be defined as: L c = A (2) where A is the area of cross-section of specimen.

The calculation result shows the d S / R T = 17.72 mm. Therefore, in the permeability test of phyllite bimrocks, any grain size larger than 17.72 mm could be considered “phyllite blocks”, and less than 17.72 mm could be regarded as “phyllite soil.” However, the standard sieves do not have 17.72 mm, and the closest size of the standard sieves less than 17.72 mm is 10 mm. Therefore, 10 mm is selected in this experimental test as the upper limit for the phyllite soil. The phyllite soil is obtained from the Maoxian Group area, and grain curves are shown in Figure 4A.

Some fine phyllite particles less than 10 mm also have the orientation of blocks in phyllite bimrocks. However, it is hard to reproduce the orientation of blocks manually due to the small particle size and colossal quantity. Besides, assuming these fine phyllite particles are used as the matrix without considering the orientation of blocks, it may affect the accuracy of the permeability test of the phyllite bimrocks. Furthermore, the orientation of blocks is the focus of this study, and the irrelevant influence factors should be excluded as much as possible. Thus, the sand is selected as the matrix because it has no orientation of blocks. Accordingly, the sand replaces the phyllite soil with the same grain size distribution (Figure 4B). According to DL/T5356-2006, the ratio between the diameter of the sample and the maximum particle size should be greater than 5. Therefore, the maximum size of phyllite blocks is 80 mm, and phyllite block sizes of 20–40, 40–60, and 60–80 mm are adopted in the test (Figures 4C–E).

After several attempts, the result indicates that the maximum value of phyllite block percentage is 35% by mass. When the phyllite block percentage is larger than 35%, the phyllite blocks not be surrounded by the sand matrix. The phyllite blocks play the role of the skeleton in phyllite bimrocks, and the holes will be generated between the phyllite blocks, which affects the accuracy of the test. Consequently, the phyllite block percentages of 35%, 25%, 15%, and 5% are adopted in this test (Figures 4E–H), in which the phyllite block size is 60–80 mm.

Two groups of phyllite bimrock specimens with different block percentages and block sizes were produced to investigate the effects on the flow behavior of phyllite bimrocks. Group G-1 considered the effects of block percentage, and four block percentages were designed, i.e., 35%, 25%, 15%, and 5%. These four specimens have the same block size of 60–80 mm. Group G-2 considered the impact of block size, and three block sizes were designed, i.e., 20–40, 40–60, and 60–80 mm. These three specimens have the same block percentage of 35%. Besides, the orientation of blocks was considered in the permeability tests. The orientation angle (α) of 30° was adopted, e.g., the orientation angle (α) of all phyllite blocks in phyllite bimrock specimens is the same.

Before preparing the specimen, the compaction degree must be considered to ensure that the sand matrix in all specimens had the same porosity. According to the Standard for Soil Test Method (GB/T50123-2019), the optimum moisture content of the sand matrix is obtained by using the proctor compaction test method. The results show that the optimum moisture content of the sand matrix is 8%, and the maximum dry density ( ρ max ) is 1.980 g/cm³. The compaction degree is calculated by the following equation: K = ρ ρ max × 100 % (3) where K is the compaction degree; ρ is the compaction density and ρ max is the maximum dry density.

In this paper, K = 80 % is adopted. Therefore, ρ is 1.587 g/cm³. The porosity of the sand matrix can be calculated as: e = G s ρ − 1 (4) n = e e + 1 (5) where G s is the specific gravity of sand matrix; e is the porosity ratio of sand matrix; ρ is the compaction density of sand matrix and n is the porosity of sand matrix. Also, according to the Standard for Soil Test Method (GB/T50123-2019), the specific gravity of sand matrix ( G s ) is obtained using the pycnometer test method, which is 2.687 g/cm³. The calculation result shows n = 0.409.

To calculate the mass of the phyllite blocks and sand matrix in specimens, several parameters of phyllite bimrock specimens are defined in this paper (see Figure 5). It is well known that soil mass is generally referred to as a three-phase system consisting of solid particles, water, and air. Furthermore, the solid particles are composed of phyllite blocks and sand particles in phyllite bimrock specimens (Figure 5). Therefore, the mass of phyllite blocks and sand matrix can be calculated by the following equations: { V p + V s w = V m p m s = ρ p ⋅ V p ρ s ⋅ V s = q V v V s w = n (6)

From Eq. 6, the mass of the phyllite blocks and sand matrix can be given as follows: { m s = ρ s ⋅ ρ p ⋅ V ⋅ ( 1 − n ) ( 1 − n ) ⋅ k ⋅ ρ s + ρ p m p = q ⋅ ρ s ⋅ ρ p ⋅ V ⋅ ( 1 − n ) ( 1 − n ) ⋅ q ⋅ ρ s + ρ p (7) where V p is the volume of phyllite blocks; V s w is the sum of the volume of water, air and sand particles; V v is the volume of void; V s is the volume of sand particles; V is the volume of phyllite bimrock specimen; m s is the mass of sand particles; m p is the mass of phyllite blocks; q is the ratio of m p and m s ; n is the porosity of sand matrix; ρ s is the density of sand particles, which is equal to the G s ; ρ p is the density of phyllite blocks (2.858 g/cm³).

From Eq. 7, the mass of phyllite blocks and sand matrix with different block percentages can be calculated. The calculation results are listed in Table 1. To achieve the orientation of blocks in phyllite bimrock specimens, the specimen is divided into five layers in vertical, and each layer is 8 cm high. The schematic diagram of the specimens with different block percentages and block sizes is shown in Figure 6. Since the specimen is divided into five layers in vertical, the phyllite blocks and sand matrix are also divided into the same five parts in each specimen. The specimen preparation process is shown in Figure 7.

Figure 7A shows the preparation process of layer 5 with a block percentage of 35% and the orientation angle (α) of 30°. The preparation process of layers 1–4 is similar to layer 5. Initially, a geotextile is placed on the perforated plate to prevent the sand particles from falling into the bottom of the seepage cylinder (Step 1). Then some sand particles are used to make a slope of 30° (Step 2), and several phyllite blocks are placed on the slope surface (Step 3). Finally, the phyllite blocks are covered with sand particles and then make a slope of 30° again (Step 4). Repeat steps 1–4 until the phyllite blocks are filled (Step 5–11), and then phyllite blocks are compressed to 8 cm in vertical (Step 12). Repeat steps 2–12, and the phyllite bimrock specimens can be prepared (Figure 7C). The preparation process of phyllite bimrock specimens with block percentages of 5%, 15%, and 25% is the same as the 35%. The specimen preparation process of different blocks size is similar to phyllite bimrock specimens with a block percentage of 35%, except that the phyllite blocks are replaced by the corresponding size. The phyllite bimrock specimens with different block sizes are shown in Figure 7B. All the phyllite blocks have the same orientation angle (α) of 30°, which is the special structure of phyllite bimrocks and is different from the other bimrocks.

During the permeability test, the experimental steps are as follows:

1) Specimen saturation. After the preparation of the specimen, the water was injected to saturate the specimen. Each specimen was soaked for 3 h.

The hydraulic gradient can be calculated by the following equation: i = Δ h L (9) where i is the hydraulic gradient; Δ h is the water head difference; L is the flow distance, which is equal to the height of the specimen.

To investigate the flow behavior of phyllite bimrocks, the Reynolds number (Re) is calculated in this paper. The Reynolds number (Re) can be calculated as: R e = v d n υ (10) where Re is the Reynolds number; v is the flow velocity; n is the porosity of the specimen; υ is the kinematic viscosity, d is the characteristic length of the specimen.

In practice, Reynolds number (Re) can be used to judge the flow behavior of the porous media obeying Darcy’s law or non-Darcy’s law. Bear (1972) reported that if the value of Reynolds number (Re) is between 1 and 10, which is calculated by the average grain size ( d ¯ ) , Darcy’s law is available. On the contrary, when the Reynolds number (Re) is greater than 10, it shows the non-Darcy’s law. As for the average size ( d ¯ ) of the non-uniformity material, there are some researchers claim it can be calculated as (Herrera and Felton, 1991): d ¯ = ∑ i = 1 n d i ⋅ w i ∑ i = 1 n w i (11) where d ¯ is the average size of the non-uniformity material; d i is the mean grain size of i th sized particles caught by averaging the sieve opening of two succeeding sieves; w i is the mass percent of the i th sized particles; i is i th sized particles obtained between two succeeding sieves. The average size ( d ¯ ) of phyllite bimrock specimens with different block percentages and block sizes are shown in Table 2.

As shown in Eq. 10, n is the porosity of the specimen, which is equal to the porosity of the phyllite bimrock specimens ( n P − S R M ) in this test. Based on Figure 5, the porosity of the phyllite bimrock specimens ( n P − S R M ) can be written by the following equation: n P − S R M = V ρ s ρ p − m s ρ p − m p ρ s ρ s ρ p V (12) where n P − S R M is the porosity of phyllite bimrock specimens; V is the volume of phyllite bimrock specimens; m s is the mass of the sand particles; m p is the mass of the phyllite blocks; ρ s is the density of sand particles, which is equal to the G s ; ρ p is the density of the phyllite blocks. The calculated results are listed in Table 2.

From Eq. 10 and Table 2, the range of Reynolds numbers has been obtained. The relationship between the Reynolds number and hydraulic gradient is shown in Figure 8.

Figure 8 shows that when block percentages are 35%, 25%, and 5% and block sizes are 60–80 and 40–60 mm, the non-Darcy’s flow is evident. Moreover, with the increase of the hydraulic gradient, the Reynolds number increases, and the flow regime becomes laminar to turbulent under a lower hydraulic gradient. However, when the block percentage is 15%, and the block size is 20–40 mm, it cannot be seen that the flow behavior is Darcy’s law or non-Darcy’s law. Generally speaking, Figure 8 illustrates that Darcy’s law can not apply to analyze the flow behavior of the phyllite bimrocks.

To further study the flow behavior of the phyllite bimrocks, the relationship between hydraulic gradient and flow velocity is studied. In previous studies, if the hydraulic gradient exists a positive correlation with the flow velocity, the flow behavior of porous media obeys Darcy’s law. However, if the hydraulic gradient has a non-linear relationship with the flow velocity, the flow behavior of porous media is regarded as the non-Darcy’s flow. In many publications, two empirical equations are frequently applied to analyze the flow behavior of porous media, which are the Forchheimer equation and the Izbash equation (Venkataraman and Rao, 1998; Bordier and Zimmer, 2000; Yamada et al., 2005; Sidiropoulou et al., 2007).

Forchheimer equation is known as quadratic correlations between flow velocity and hydraulic gradient. Forchheimer (1901) added a second order velocity term to the original Darcy equation representing the microscopic inertial effect in groundwater flow (Hansen, 1992), and the Forchheimer equation can be written as: i = A v + B v 2 (13) where A and B are called the Forchheimer coefficient; i is the hydraulic gradient; v is the flow velocity.

It is assumed that the power relationship can be established by fitting the hydraulic gradient with the flow velocity. In that case, the Izbash equation can be used to evaluate the flow behavior of porous media, and the Izbash equation can be written as: i = a v b (14) where a and b are called the Izbash coefficient; i is the hydraulic gradient; v is the flow velocity.

To evaluate the flow behavior of phyllite bimrocks, the regression relationship of the hydraulic gradient and the flow velocity of phyllite bimrock specimens with different block percentages and block sizes considering the orientation of blocks are depicted in Figures 9, 10. Figure 9 shows i—v correlation using quadratic law (Forchheimer equation), and Figure 10 shows the power regression of i—v correlation (Izbash equation).

Figure 9 shows that the quadratic relationship can be established by fitting the hydraulic gradient with the flow velocity. The quadratic curve (Forchheimer equation) is in good agreement with the experimental data. Therefore, the flow behavior of phyllite bimrocks with different block percentages and block sizes considering the orientation of blocks does not obey Darcy’s law. Besides, compared with Figure 8, when the block percentage is 15% and the block size is 20–40 mm, the relationship between the hydraulic gradient and the flow velocity is not linear obviously. Therefore, it is clear that even though the Reynolds number is smaller, the flow behavior of phyllite bimrocks does not comply with Darcy’s law. Figure 10 shows that the power relationship (Izbach equation) can be established by fitting the hydraulic gradient with the flow velocity. To get a satisfactory result, the Forchheimer coefficients (A and B), Izbash coefficients (a and b) and the coefficient of regression (R ²) are calculated, and the results are shown in Table 3.

It can be seen from Table 3 that whether the Forchheimer equation or Izbash equation, the experimental data has a good correlation. However, R ² obtained by fitting the Forchheimer equation is larger than 0.995, while R ² obtained by fitting the Izbach equation is less than 0.995 (except blocks percentage = 25%). Therefore, compared with the Izbach equation, the Forchheimer equation can be better used to analyze the relationship between hydraulic gradient and the flow velocity of the phyllite bimrocks.

In summary, the presented results show that Darcy’s law is not available to determine the flow behavior of phyllite bimrocks with different block percentages and block sizes considering the orientation of blocks, but it can be better analyzed by the Forchheimer equation.

As mentioned before, the Forchheimer equation gives more satisfactory results relative to the Izbash equation. Thus, in this section, the hydraulic conductivity of phyllite bimrocks with different block percentages and block sizes considering the orientation of blocks is calculated using the Forchheimer equation. For the Forchheimer coefficient A, Zeng and Grigg (2006) evaluate by the following expression: A = μ ρ g k (15) where A is the Forchheimer coefficient; μ is the coefficient of dynamic viscosity of fluid; ρ is the density of fluid; g is the gravitational acceleration and k is the intrinsic permeability.

In addition, the hydraulic conductivity ( k s ) can be expressed as the following equation (Zanker, 1972): k s = ρ g k μ (16) where k s is the hydraulic conductivity; μ is the coefficient of dynamic viscosity of fluid; ρ is the density of fluid; g is the gravitational acceleration and k is the intrinsic permeability.

From the Eqs 15, 16, the following equation can be obtained: k s = 1 A (17) where k s is the hydraulic conductivity and A is the Forchheimer coefficient.

Based on Eq. 17 and Table 3, the hydraulic conductivity ( k s ) can be calculated, and the calculation results are illustrated in Figure 11.

Figure 11A plots hydraulic conductivity against block percentage considering the orientation of blocks. With the increase in block percentage, the hydraulic conductivity firstly decreases to the minimum and then increases, and the minimum of hydraulic conductivity is the percentage of the block of 25%. The result is in disagreement with Dunn and Mehuys (1984) and Liao (2004). Their research shows that the hydraulic conductivity constantly decreases with the increase in the rock blocks. The inconsistency of the result may have been due to the relative waterproof of phyllite blocks and the permeability of block-sand interfaces. Compared with the sand matrix, the phyllite blocks are relatively waterproof and reduce the effective flow cross section, which means the effective porosity in phyllite bimrocks decreases. Therefore, the porosity of phyllite bimrock specimens decreases (see Table 2), resulting in the hydraulic conductivity decreasing with the block percentage increase. Nevertheless, because the phyllite blocks are relatively waterproof, when the seepage water flows through the phyllite bimrocks, it prefers to flow along the surface of phyllite blocks rather than in the sand matrix. The hydraulic pressure decreases sharply at the block-sand interfaces, and the permeability of block-sand interfaces increases accordingly. Thus, the permeability property of phyllite bimrocks increases as increasing the block percentage. Consequently, combined with the two influencing factors, the result shows that the hydraulic conductivity of the phyllite bimrocks firstly increases and then decreases. When the block percentage is less than 25%, the relative waterproof of phyllite blocks is dominant, and the hydraulic conductivity decreases with the increase of the block percentage. However, when the block percentage is larger than 25%, the high permeability of block–sand interfaces leads to increased hydraulic conductivity as the block percentage increases.

During the permeability test conducted by Wang et al. (2016a), the trend of block percentage against hydraulic conductivity is roughly similar to the result in this paper. However, Wang et al. (2016a) presented that the hydraulic conductivity reaches the minimum when the block percentage is 40%. The reason why the results are different is that the orientation of blocks is ignored. Compared with the homogeneous bimrocks (Figures 11C,D), the flow paths in the phyllite bimrocks decrease due to the orientation of blocks. The orientation of blocks further enhances the permeability property of block-sand interfaces, and the hydraulic conductivity reaches its minimum when the block percentage is smaller.

Figure 11B shows that hydraulic conductivity increases approximately linear with the increase in block size. In this paper, the relationship between hydraulic conductivity and block size is contrary to Shafiee (2008) and Gutierrez and Vallejo (2013). The reason for the difference is that the orientation of blocks is considered in this paper. As mentioned earlier, the permeability of block-sand interfaces is larger than the sand matrix. When the block size increases, the permeability of block-sand interfaces increases accordingly. Besides, the orientation of blocks reduces the actual flow paths in the phyllite bimrocks. Under the combined effect of larger permeability of block-sand interfaces and shorter flow path, the permeability property of phyllite bimrocks has been significantly enhanced. Therefore, the hydraulic conductivity of phyllite bimrocks goes up with the increase in block size.

Based on the results of section 3.2, it was found that the flow behavior of the phyllite bimrocks does not comply with Darcy’s law and the Forchheimer equation can be better used to analyze the flow behavior of the phyllite bimrocks. In 1901, Forchheimer presented the quadratic equation after the experimental test. However, the effects of the physical and mechanical properties of porous media on Forchheimer coefficients A and B have not been studied. After that, many scholars and engineers are interested in determining the Forchheimer coefficient A and B. For the A and B, Ergun (1952) suggested the following expressions: A = 150 υ ( 1 − n ) 2 g n 3 d 2 , B = 1.75 ( 1 − n ) g n 3 d (18) where A and B are the Forchheimer coefficient; n is the porosity of the porous media; d is the particle size; g is the gravity acceleration and υ is the kinematic viscosity of the fluid.

Kovacs (1981) derived the following expressions of Forchheimer coefficient A and B: A = 144 υ ( 1 − n ) 2 g n 3 d 2 , B = 2.4 ( 1 − n ) g n 3 d (19) where A and B are the Forchheimer coefficient; n is the porosity of the porous media; d is the particle size; g is the gravity acceleration and υ is the kinematic viscosity of the fluid.

Fand and Thinakaran (1990) present the following expressions of Forchheimer coefficients A and B: A = 214 M 2 ( 1 − n ) g n 3 d 2 , B = 1.57 M ( 1 − n ) g n 3 d M = 1 + 2 3 d D ( 1 − n ) (20) where A and B are the Forchheimer coefficient; n is the porosity of the porous media; d is the particle size; g is the gravity acceleration; M is a function of A and B and υ is the kinematic viscosity of the fluid.

Furthermore, a similar approach of Forchheimer coefficient A and B has also been suggested by Kadlec and Knight (1996): A = 255 υ ( 1 − n ) g n 3.7 d 2 , B = 2 ( 1 − n ) g n 3 d (21) where A and B are the Forchheimer coefficient; n is the porosity of the porous media; d is the particle size; g is the gravity acceleration and υ is the kinematic viscosity of the fluid.

The above expressions for A and B are representative examples in the publication. It can be seen that the Forchheimer coefficient A and B have closely relevant to the particle size and porosity. As mentioned before, block percentage is closely related to the porosity of the phyllite bimrocks ( n P − S R M ) , and the average size ( d ¯ ) (see Table 2). In this section, the particle size (d) is difficult to determine due to the particle size distribution of the sand matrix and the phyllite block size is not constant. Thus, the average size ( d ¯ ) is adopted to represent the particle size (d).

Accordingly, based on the Eqs 18–21, Forchheimer coefficients A and B are defined by the following expressions in this paper: A = A 1 υ ( 1 − n P − S R M ) 2 g n P − S R M d ¯ 2 3 , B = B 1 ( 1 − n P − S R M ) g n P − S R M d ¯ 3 (22) where A ₁ and B ₁ are the constant value. A and B are the testing values (Table 3); n P − S R M is the porosity of phyllite bimrock specimens; d ¯ is the average size; g is gravity acceleration and υ is the kinematic viscosity of the fluid.

To determine the Forchheimer coefficients A and B, the normalized objective function (NOF) criterion is used in this paper. The NOF is the ratio between the Root Mean Square Error (RMSE) and the overall mean X of the testing data. The NOF, RMSE, and X can be calculated as follows (Moutsopoulos et al., 2009): N O F = R M S E X (23) RMSE = ∑ i = 1 N ( x i − y i ) 2 N (24) X = 1 N ∑ i = 1 N x i (25) where x _i are the testing values of A and B (see Table 3); y _i are the calculation values by Eq. 22; N is the total number of values. To get the best match, NOF must be less than 1.0. When the NOF is less than 1.0, the theoretical method is still reliable. The computed results are shown in Table 4.

As shown in Table 4, the calculated values of NOF(B ₁ ) are closer to 0.0 than that of NOF(A ₁ ), especially for the block size of 60–80 mm. For the block percentage, when the block percentage is 5%, both NOF(A ₁ ) and NOF(B ₁ ) are less than 1.0, which indicates that A ₁ = 988.830 and B ₁ = 3.057E6 can be used to determine the value of A and B roughly. For the block size, except the block size is 60–80 mm, NOF(A ₁ ) and NOF(B ₁ ) are less than 1.0. When the block size is 40–60 mm, the A ₁ = 1110.264 and B ₁ = 9.426E6 can be adopted to estimate the value of A and B. Therefore, Forchheimer coefficients A and B can be expressed as follows in this paper: i = 988.830 υ ( 1 − n ) 2 g n 3 d 2 V + 3.057 E 6 ( 1 − n ) g n 3 d V 2 (26) i = 1110.264 υ ( 1 − n ) 2 g n 3 d 2 V + 9.426 E 6 ( 1 − n ) g n 3 d V 2 (27) where i is the hydraulic gradient; V is the flow velocity; n is the porosity of the porous media; d is the particle size; g is the gravity acceleration and υ is the kinematic viscosity of the fluid.

Eq. 26 and Eq. 27 are suitable for describing the flow behavior of phyllite bimrocks with different block percentages and block sizes considering the orientation of blocks, respectively. In addition, it is worth noting that these two flow laws have strict applicable conditions. First, it can not be applied to other bimrocks, only the phyllite bimrocks. Second, the orientation angle of the phyllite rock must be 30°, and other orientation angles are not suitable. Third, the compaction degree must be 80%, i.e., the porosity of the sand matrix must be constant. Consequently, these two flow laws have a narrow scope of application. In the following research, these problems will be figured out.