Experimental Investigation of the Non-Darcy Equivalent Permeability of Fractured Coal Bodies: The Role of Particle Size Distribution

Abstract

The permeability of crushed coal bodies plays a bottom neck role in seepage processes, which significantly limits the coal resource utilisation. To study the permeability of crushed coal bodies under pressure, the particle size distribution of crushed coal body grains is quantitatively considered by fractal theory. In addition, the parameters of the percolation characteristics of crushed coal body grains are calculated. Moreover, the permeability of the crushed coal body during recrushing is determined by the fractal dimension and porosity. A lateral limit compression test with the crushed coal bodies was carried out to illustrate the effect of the porosity on the permeability, In addition, a compressive crushed coal body size fractal–permeability model was proposed by combination of the fractal dimension and the non-Darcy equivalent permeability. The results show (1) the migration and loss of fine particles lead to a rapid increase in the porosity of the crushed coal body. (2) Increases in the effective stress cause the porosity and permeability to decrease. When the porosity decreases to approximately 0.375, its effect is undermined. (3) The migration and loss of fine particles change the pore structure and enhance the permeability properties of the skeleton, causing sudden seepage changes. (4) At low porosity, the permeability k is slightly larger than the non-Darcy equivalent permeability ke. Thus, the experimental data show an acceptable agreement with the present model. A particle size fractal–percolation model for crushed coal bodies under pressure provides a solution for effectively determining the grain permeability of the crushed coal bodies. The research results can contribute to the formation of more fractal-seepage theoretical models in fractured lithosphere, karst column pillars and coal goaf, and provide theoretical guidance for mine water disaster prevention.

1. Introduction

Mine water damage is an important constraint in the use of coal resources and is commonly induced by karst column pillars (KCP) [1,2], which contain a considerable amount of broken coal grains [3]. To improve the efficiency of coal resource utilisation and ensure the efficient extraction of coal and gas energy, mine water damage systems must be thoroughly investigated. Therefore, the characteristic permeability parameters of crushed coal body systems should be studied to provide a theoretical understanding for managing mine water damage by improving a calculation method for the crushed coal body grain permeability system, thus ensuring the safe and efficient extraction of coal resources.

The permeability of broken coal body grains is used to quantitatively characterise the ease of fluid passage in porous media [4]. The percolation properties of fractured coal bodies have been investigated through numerous percolation tests [5,6,7]. Mckee et al. [8] concluded that effective stresses can fracture rock, affecting the permeability, porosity and density. The seepage behaviour of fractured coal bodies exhibits pronounced nonlinear flows [9,10], and Moutsopoulos et al. [11] investigated the flow processes that occur on the pore scale for non-Darcy flows. Moreover, Yao et al. [12] indicated that the non-Darcy parameters (permeability k, non-Darcy flow β coefficient, etc.) of fractured rock masses are related to the porosity. Ma et al. [13] tested the seepage characteristics of fractured rock masses using the MTS815.02 test system and determined the permeability k in the steady state. Shi et al. [14] analysed the porosity evolution process and obtained the relationship between the permeability k and porosity. Sterpi et al. [15] and Cividini et al. [16] developed a particle size-based finite difference model and analysed how the particle size affects the seepage characteristics. Pang et al. [17] established the functional relationship between the permeability and fractured rock grain size characteristics. Kong et al. [18] analysed variations in the permeability and porosity of fractured rock based on variations in the particle size distribution with the non-Darcy percolation genetic algorithm.

Despite the above advances, the above methods generally ignore the granularity distribution of the fractal angles. Under pressure, the influence of particle size is prominent, which makes it difficult to predict the permeability. Therefore, the permeability of the fractal coal mass is studied based on fractal theory, and the calculation model is proposed as the solution method.

In this paper, the permeability parameters of the different pore structures are analysed and calculated based on the particle size distribution of crushed coal, and the influence of porosity on permeability is elaborated in detail. By connecting the fractal dimension with the non-Darcy equivalent permeability, the fractal permeability model of crushed coal is established, and the usability of the model is verified. The research results can provide theoretical guidance for mine water disaster prevention and control.

2. Theory

In a medium with a considerable amount of broken coal grains, percolation is typically nonlinear, and a prerequisite for studying the permeability of this type of medium is the ability to accurately and quantitatively describe its particle distribution [19].

(1)

Principle of the particle size distribution of crushed coal bodies

The size distribution, deformation and damage characteristics of bulk (heap solid) structures have long been studied, and Tyler, a leading scholar in the field of fractals, has found through extensive research that the size distribution has fractal characteristics during the recrushing of compressed crushed coal bodies. Therefore, fractal theory should be used to study the fractal characteristics of the particle size distribution and compression deformation properties during the recrushing of compressive crushed coal bodies. The cylinder used to load the crushed specimens in this test is quenched, and its strength and stiffness fully satisfy the test requirements, such as no radial deformation.

According to previous studies on the fractal characteristics of crushed rocks, the ratio of the mass of crushed particles with scales less than d to the mass of the whole specimen is [20]:

$$\frac{M_{\mathrm{d}}(x<d)}{M_{\mathrm{t}}}=\frac{d^{3-D}-d_{\min }{}^{3-D}}{d_{\max }{}^{3-D}-d_{\min }{}^{3-D}}$$

(1)

Assuming that the minimum particle scale d_min in a crushed specimen is 0, Equation (1) can be simplified to the following form [21]:

$$\frac{M_{\mathrm{d}}(x<d)}{M_{\mathrm{t}}}={(\frac{d}{d_{\max }})}^{3-D}$$

(2)

where M_d is the mass of the particles in the crushed specimen with scales less than d; M_t represents the sum of the masses of all the particles in the specimen; d is a particular particle scale; d_max is the maximum scale in the medium; and D is the fractal dimension of the crushed particle size.

Taking the logarithm of both sides of Equation (2) yields [22]:

$$\mathrm{l}\mathrm{g}{M}_{\mathrm{d}}-\mathrm{l}\mathrm{g}{M}_{\mathrm{t}}=\left(3-D\right)\cdot \left(\mathrm{l}\mathrm{g}d-\mathrm{l}\mathrm{g}{d}_{\max }\right)$$

(3)

Based on the data measured in the test and Equation (3), a straight line can be fitted on the double logarithmic axis with a slope of a = 3 − D, which in turn gives the fractal dimension D of the particle size of the crushed specimen.

(2)

Non-Darcy percolation principle in crushed coal bodies

The porosity is an important parameter that affects the seepage characteristics of the fractured rock and is calculated as follows [23]:

$$\varphi =\frac{V-V_0}{V}$$

(4)

where $\varphi$ is the porosity; V is the volume of the specimen after crushing, m³; and V₀ is the volume of the intact specimen, m³.

The pressure gradient of the fluid can be calculated according to the difference in the osmotic pressure between the two ends of the crushed specimen as follows [24]:

$$\frac{\partial P}{\partial x}=\frac{P_2-P_1}{H}$$

(5)

where P₁ is the pore pressure when the fluid flows into the specimen, MPa; P₂ is the pore pressure when the fluid flows out of the specimen and has a value of 0 because of the direct air connection at the outlet end of the percolation (ignoring the effect of gravity); and H is the height of the broken specimen in the percolation test, m.

Numerous experimental studies have shown that the seepage pattern of fractured rock masses satisfies Forchheimer’s empirical formula [25], namely:

$$-\frac{\partial P}{\partial x}=\frac{\mu }{k}v+\rho \beta v^2$$

(6)

where ∂P/∂x represents the pressure gradient of the fluid at the top and bottom surfaces of the specimen; v is the velocity of the fluid flowing in the specimen, m/s; µ is the dynamic viscosity of the fluid, Pa·s; ρ is the density of the fluid, kg/m³; k is the permeability, m²; and β is the non-Darcy flow factor, m⁻¹.

3. Experiments

The Talbot power index n controls the degree of compactness and the particle size distribution of the crushed coal body, both of which indirectly reflect the porosity and fractal dimension of the medium. Therefore, we use different initial gradations to control the initial porosity and fractal dimension and change the compactness and particle size distribution by loading to analyse the effect on the permeability.

3.1. Materials

(1)

Research background

The samples were obtained from the He yang Mine of the Cheng he Mining Bureau in Shaanxi Province, which is located in the eastern part of the Wei bei coalfield. The coal-bearing strata in this mine are of Carboniferous–Permian age, and the No. 5 coal seam, which was mined at an elevation of +321~+340 m, was the main coal seam. The No. 5 coal seam is stable, has a hardness that ranges from 0.5 to 1, and is a typical “three soft” coal seam.

(2)

Material preparation

To prepare the crushed rock specimens, larger pieces of coal were crushed with a CP-330 hammer crusher. Then, the five different particle sizes that were used in the test, namely, 0~5 mm, 5~10 mm, 10~15 mm, 15~20 mm and 20~25 mm, were screened with a BZS-200 standard vibrating sieve, as shown in

Table 1. Initial masses of the crushed specimens for each particle size interval.

Grain Size/mm	0~5	5~10	10~15	15~20	20~25
Mass/g	220.76	163.60	147.27	137.58	130.79

.

Finally, the mass of each group of rock samples was set to 800 g according to the volume of the cylinder. The masses of the specimens in each grain size interval were proportioned according to the Talbot formula [26], and the mass distribution of each initial graded specimen is shown in Table 1.

$$P={\left(\raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$d_{\max }$}\right.\right)}^n\times 100\%$$

(7)

where P is the proportion of particles in the crushed sample that passed through the sieve; d is the diameter of the sieve, mm; d_max is the maximum particle size in the crushed coal sample, mm; and n is the value of the Talbot power index. The crushed specimens in the different particle size intervals are shown in Figure 1 and Figure 2.

(3)

Experimental procedure

To study the seepage characteristics and particle size distributions of the crushed coal body under different initial gradations and axial stresses, specimens with four different gradations were prepared (selected to be representative of the site, with Talbot power indices n of 0.2, 0.4, 0.6 and 0.8). Five sets of specimens were prepared for each ratio, and each set of specimens was evaluated at different stress levels (determined according to the coal seam burial depth and mining disturbances σ_ov of 2, 4, 8, 12 and 16 MPa) and permeability pressures (P of 0.5, 1.0, 1.5, 2.0 and 2.5 MPa). The steady-state permeation method was used in this test. The permeate was composed of water; the density ρ of water at room temperature is 1.0 × 10³ kg/m³ and the dynamic viscosity µ is 1.01 × 10⁻³ Pa·s. The specific test procedure can be divided into five steps:

①

The prepared specimen is loaded into a cylinder that is lined with a permeable sheet and felt to prevent the loss of small coal grains.

②

A piston with a seal is fitted into the cylinder, a penetrometer is placed on the operating table of the press and the press is adjusted until it meets the piston; to ensure adequate contact, the actual stress on the specimen at this point is approximately 0.01 MPa. Then, the initial height of the specimen is measured.

③

An axial load is applied to the specimen by a computer-controlled system and a high-pressure pump is switched on when the load reaches the set stress level. Then, the percolation test is completed and the flow rate is recorded at five successively increasing levels of osmotic pressure.

④

The specimens are removed from the cylinder, dried, sifted and weighed, and the fractal dimension is calculated.

⑤

At the end of the test, the same steps are carried out with the next set of specimens until all tests are completed.

3.2. Methodologies

(1)

Experiment design

In this paper, uniaxial lateral limit compression tests were carried out with the crushed coal bodies. The test system includes a DDL600 electronic universal testing machine, a crushed rock bearing device and a computer data acquisition system, as shown in Figure 3. The crushed rock bearing device includes an indenter, a piston, a base and a cylinder of size Φ 100 mm × L 200 mm. A schematic diagram of the uniaxial lateral confinement compression and seepage in a crushed coal body is shown in Figure 3.

(2)

Experimental equipment

The fractured rock percolation test rig used for this test is shown below and consists of a press, computer, high-pressure pump and permeameter.

The permeameter consists of a cylinder, a piston and a permeation plate. The cylinder, which has a diameter of 10 cm and a height of 22 cm, is used to load the broken specimen. The computer controls the DDL600 electronic universal testing machine to apply the axial loads to the broken specimen through the permeameter and collects the raw data obtained during the loading process in real time. The variable permeation pressure and variable osmotic pressure in the test are provided by a high-pressure water pump, and the water pump control cabinet has sensors that displays the percolation flow in real time. A sample installation diagram is shown in Figure 4.

4. Results and Discussion

The permeability characteristics of the crushed coal medium, which includes a considerable amount of crushed coal body grains, are affected by the particle size fragmentation degree and pore structure deformation. Therefore, the authors use the particle size distribution of the crushed coal body to calculate the corresponding permeation parameters and analyse the permeation behaviour and the effect of several factors, such as the effective stress and the porosity, on the non-Darcy equivalent permeability of the crushed coal body. Based on the above results, the authors propose a particle size fractal–permeation model for the pressurised crushed coal bodies and verify the practicality of the model with the above experimental data.

4.1. Basic Characteristics of Infiltration Flow Variations in Seepage Systems

The variation in the percolation flow parameters can visually reflect the flow conditions of the fluid in the crushed coal body. The data collected by the paperless recorder can be used to plot the time-varying curves of the percolation flow rate for specimens with different initial gradations. For example, the time-varying curves at an axial compression displacement of 5 mm are shown in Figure 5.

Figure 5 shows that with increasing seepage time, specimens A-1 to A-4 all underwent sudden changes in seepage flow. Based on the seepage flow trend, the variable mass seepage process of the crushed coal body can be divided into three stages: (1) the initial seepage stage; (2) the abrupt seepage stage; and (3) the steady-state seepage stage. The difference in the initial gradation caused the crushed specimens to spend different amounts of time in the three percolation stages. Specimen A-1 did not undergo the initial percolation stage and instead experienced a direct abrupt change in the percolation, with the flow rate increasing to 260.1 L/h immediately; then, the specimen entered the steady-state percolation stage, where only small fluctuations in the flow rate were observed. The other three specimens experienced all three phases. Specimens A-2 and A-4 underwent sudden changes in seepage at approximately t = 59 s and t = 95 s, respectively, with instantaneous flow rates of 244.5 L/h and 163.1 L/h. The flow rate of the specimens decreased slightly before the specimens entered the steady-state seepage phase. Specimen A-3 experienced a sudden change in seepage at approximately t = 77 s, and the flow rate increased to 161.3 L/h. The flow rate of the specimen decreased for 2 s before the sudden seepage change.

The reason for this phenomenon is that, at the beginning of the seepage, water scouring causes changes in the skeletal structure of the specimen. At this time, the pores are penetrated and fine particles are jammed into the pores, and a stable seepage channel has not yet been formed; thus, the seepage flow rate fluctuates to some extent. After the sudden change in the seepage flow, the skeletal structure and seepage channels are essentially stable because a large number of fine particles are lost, and few freely migrating filled particles remain in the specimen. Thereafter, the steady-state phase of variable mass percolation is entered and the flow rate fluctuates only within a small range.

4.2. Time-Varying Characteristics of the Porosity for Structures with Different Initial Gradations

The porosity is the main parameter that characterises the pore structure of a crushed body and is an important indicator of the seepage characteristics. The initial porosity φ₀ of a crushed specimen before infiltration can be calculated with the following equation:

$${\varphi }_0=\frac{V_0-V_z}{V_0}=1-\frac{M}{\pi a^2{h}_0{\rho }_s}$$

(8)

where V₀ is the volume of the crushed specimen after compression, m³; Vz is the volume of the intact specimen, m³; M is the mass of the crushed specimen before penetration, kg; a is the radius of the penetrometer, m; h₀ is the height of the crushed specimen after compression, m; and ρ_s is the density of the intact specimen, kg/m³.

During permeation, the loss of fine particles changes the pore structure and porosity of the crushed specimen. The porosity of the crushed specimens in the permeation test can be calculated as:

$${\varphi }_i=1-\frac{M-\left({\displaystyle \sum _{i=1}^n\Delta m_i}\right)}{\pi a^2{h}_0{\rho }_s}\left(i=1,2,3,\dots ,n\right)$$

(9)

where ∆m_i represents the loss in particle mass in the crushed specimen at a specific moment in time. The rate of change in the porosity of a crushed specimen during infiltration can be calculated as:

$${\varphi }_i{}^{\prime }=\frac{{\varphi }_i-{\varphi }_{i-1}}{\Delta t}$$

(10)

where ∆t is the interval time, s.

(1)

Time-varying characteristics of the porosity

The variation in the porosity can be controlled by axial compression, which can be calculated based on the data for different axial displacements.

Based on the porosity data of the four initial grades of crushed specimens designed for the test at four levels of axial deformation, the porosity variation curve with time during percolation can be plotted using Origin plotting software, as shown in Figure 6.

As seen from the figure, for the four initial gradations and different axial deformations, the porosity of the crushed specimen has the same change trend with the seepage time; as the seepage time increases, the deformation shifts from rapid increases to smooth fluctuations. With the migration and loss of fine particles, the porosity of the broken specimens increases; that is, the process underlying the mass loss of the broken specimens is the same process underlying the porosity increase. After a large number of fine particles are lost with the fluid, the porosity remains essentially unchanged.

As an example, consider an axial compression of 5 mm, as shown in Figure 6a. From the longitudinal perspective, the porosity of specimen A-1 changed the most, increasing from an initial porosity of 0.3924% to 0.5109%, or an increase of 11.8%. The porosity of specimen B-1 changed the least, by only 0.4%, while the porosities of specimen C-1 and specimen D-14 increased from 0.4121% to 0.4946% and 0.4253% to 0.4628%, respectively, representing increases of 8.3% and 3.8%, respectively. Among the specimens, specimen A-1 and specimen C-1 had the fastest increases from t = 0 to 30, while specimen D-1 had the fastest increase from t = 90 to 120 s. The porosity increased slower in the later period, indicating that the mass loss rate gradually decreased.

(2)

Basic variation characteristics of the rate of change in the porosity

The rate of the porosity change reflects the rapidity of pore changes and indirectly reflects the mass loss of the crushed specimens. Origin plotting software was used to plot the variation curves of the porosity change rate with the percolation time for each specimen under different axial compression displacements, as shown in Figure 7.

According to the magnitude of the slope of the time-varying curve of the porosity change rate of the specimen, the change process can be divided into two stages: (1) the rapid change stage and (2) the slow change stage. Consider an axial deformation of 5 mm as an example, as shown in Figure 7b. Specimen A-1 has a complex change trend in the time-varying curve before t = 135 s, with both increases and decreases; however, after this time, the curve changes with smaller increases and decreases. The rate of porosity change of specimens B-1, C-1 and D-1 shows a maximum value at t = 105 s, after which the rate of porosity change shows a monotonically decreasing trend. In the early stage of seepage, the fractured particles are prone to rotating, rolling and sliding due to the scouring effect of the water flow; as a result, the loss of fine particles at the boundary is large during this time period and the porosity change rate fluctuates significantly.

As the seepage time increases, the permeability meter gradually loses all the fine particles, which can migrate freely, leaving only the solid skeleton part and the filler particles, which have difficulty migrating. The water has certain chemical corrosion and physical dissolution effects on the broken coal body; however, significant effects are unlikely during very short seepage processes. Therefore, secondary powder-like particles are produced and mixed in the fluid only when the water-rock action is longer, forming a suspension that flows out of the permeameter with the fluid, with a gradual increase in the porosity and a gradual decrease in the rate of change in the porosity.

(3)

Pore structure changes with different initial gradations

When the specimen is compressed axially under compression, the skeleton of the broken specimen will deform to some extent. The pore structure is adjusted, which may increase or decrease. It can be reflected by the change in the rate of change of porosity. According to the calculation results, the histogram of the change of the porosity change rate of each sample with the seepage time under different initial gradations is drawn, as shown in Figure 8.

It can be seen from Figure 8 that, taking the initial gradation n = 0.2 as an example, the porosity change rate of each broken sample under different gradation conditions generally decreases with the increase in the axial displacement, and a sudden change occurs locally. Among them, compared with other broken samples in group A, sample A-1, that is, the axial deformation amount is 5 mm, and the porosity change rate is the maximum after t = 15 s. When t = 15 s, the axial deformation of sample A-2 is 10 mm and the porosity change rate is the maximum value. The maximum is 14.08%. The reason is that when the axial displacement is 5 mm, the structure of the sample is relatively loose, and a large number of fine particles are easy to migrate under the erosion of water, and then lose from the boundary, resulting in the largest porosity change rate. Then in the process of increasing axial displacement, the coal particles are further broken, and the pore structure is continuously adjusted. There may be two changes: one is that the increase in fine particles inside the sample blocks the seepage channel, the structure of the sample is more compact and the rate of change of porosity decreases. The other is that the inside of the sample can be repositioned to generate new percolation channels and increase the rate of porosity change.

4.3. The Decisive Influence of the Porosity on the Specimen Permeability

Prior to the sudden change in seepage, the solid fine particles of the crushed specimen migrate and are lost; furthermore, their pore structures and permeabilities constantly change. However, after the sudden change in percolation occurs, as the percolation time increases, the fine particles are no longer present, the skeleton and filler particles eventually stabilise and the permeability no longer changes. At this time, by gradually reducing the osmotic pressure, the percolation velocity of the crushed specimen under different osmotic pressures can be obtained. The pore pressure gradient and percolation velocity of a specimen with an initial gradation of n = 0.2 were determined under four axial displacements with Forchheimer’s formula and Darcy’s law formula, respectively. The fitted curves and fits of this specimen are shown in Figure 9.

Figure 9 shows that the nonlinear correlation coefficient between the pore pressure gradient and the seepage velocity is considerably larger than the linear correlation coefficient when the fluid flow in the crushed specimen reaches the steady-state seepage stage. This result shows that the variable mass percolation characteristics of the crushed coal body also exhibit a Forchheimer-type nonlinear percolation form. The variation in the permeability k with the porosity φ is shown in Figure 10.

The relationship between the porosity φ and permeability k for different Talbot power indices n at different axial displacements is plotted according to the test results, as shown in Figure 10. For fractured coal bodies, the most important parameter affecting the permeability is the porosity. The test data show that the porosity and permeability of the crushed specimens decrease as the axial deformation increases, with the minimum porosity reaching 0.298.

In this test study, when the porosity decreased to approximately 0.375, the permeability values of fractured specimens with different initial gradations were similar. As the porosity continued to decrease, its effect on the permeability was no longer apparent. The main influencing factor is the different specific surface areas of the crushed specimens at different particle size ratios. The arrangement and spatial distribution of the crushed coal particles are somewhat random, and the pore distribution state, pore size and formation of effective seepage channels in the specimens have some influence on the seepage characteristics.

4.4. Equivalent Permeability Mathematical Model for the Crushed Coal Seepage System

In previous studies, the relationship between the pore fractal dimension and the permeability was modelled using the pore space as the object of study; however, particles have rarely been used to model the relationship between the particle size fractal dimension and the permeability. A study of the particle distribution and seepage characteristics of crushed coal bodies shows that the porosity and fractal dimension of the particle size have significant effects on the permeability. Therefore, in this section, the fractal dimension of the particle size is related to the non-Darcy equivalent permeability based on the spherical particle and capillary assumptions to establish a mathematical model between the two parameters.

The non-Darcy flow in porous media mainly changes with the flow velocity, which also leads to the deviation between the pressure gradient and the movement velocity, and the fluid shows non-Darcy flow characteristics. Permeability is a fundamental parameter that describes the ease of fluid passage through porous media. Various scholars have used theoretical derivations and laboratory tests to demonstrate that the permeability of porous media is controlled by the pore structure. The Kozeny–Carman (KC) equation has been widely used and has the following expressions [27]:

$$k_{\mathrm{d}}=\frac{{\varphi }^3}{C{S}^2{\left(1-\varphi \right)}^2}$$

(11)

where k_d is the permeability of the crushed coal body in a Darcy percolation form, S is the specific surface area of the crushed specimen and C is the empirical constant of the KC equation, the value of which is generally taken as 5 [28]. Figure 11 shows the fractal-seepage hypothesis model of broken coal rock mass.

The specific surface area, which is denoted as S, is the surface area of all pores within a unit volume of porous media [29]. Assuming that the crushed coal particles are composed of N spheres of unequal diameters with a radius r_i, and the total surface area of the pores A_s (total surface area of the spheres) and total volume of the solid particles (total volume of the spheres) vs are:

$$A_s={\displaystyle \sum _i^{N}4\pi r_i{}^2}, V_s={\displaystyle \sum _i^N\frac{4}{3}\pi r_i{}^3}=\left(1-\varphi \right)V_b$$

(12)

where V_b is the sum of the solid volume and pore volume of the crushed coal body, i.e., the total volume, and φ is the porosity of the crushed coal body.

Then, the specific surface area can be calculated as:

$$S=\frac{A_s}{V_b}=3\left(1-\varphi \right){\displaystyle \sum _i^N{r}_i{}^2}/{\displaystyle \sum _i^N{r}_i{}^3}=\frac{3\left(1-\varphi \right)}{\overline{r}}$$

(13)

where $\overline{r}={\displaystyle \sum _i^N{r}_i{}^3}/{\displaystyle \sum _i^N{r}_i{}^2}$ is the mean radius.

The object of study in this work is a spherical particle; thus, $\overline{d}=2\overline{r}$ [30]. Then, Equation (11) can be written as:

$$k_d=\frac{{\varphi }^3}{36C{\left(1-\varphi \right)}^2}{\overline{d}}^2$$

(14)

where $\overline{d}$ is the average diameter of the spherical particles, which can be determined according to the grading curve $d_{50}$.

Equation (6) can be written in the form of Darcy’s law as follows:

$$-\nabla P=\frac{\mu }{k_e}v=\left(\frac{1}{k_d}+\frac{\rho \beta v}{\mu }\right)\mu v$$

(15)

where k_e is the non-Darcy equivalent permeability of the fractured coal body and k_d is the Darcy permeability. The non-Darcy equivalent permeability can be expressed as:

$$k_e=\frac{\mu k_d}{\mu +\rho \beta k_{d}v}$$

(16)

Many researchers have explored the non-Darcy percolation factor β in the Forchheimer equation; the most popular form is the Ergun equation, which was summarised by Ergun based on a large amount of experimental data, the theory of percolation flow and the mean hydraulic radius, which has the following expression [31,32]:

$$-\nabla P=\frac{150\mu {\left(1-\varphi \right)}^{2}v}{{\overline{d}}^2{\varphi }^3}+1.75\frac{\left(1-\varphi \right)\rho v^2}{{\varphi }^3\overline{d}}$$

(17)

where:

$$\beta =\frac{1.75\left(1-\varphi \right)}{{\varphi }^3\overline{d}}$$

(18)

According to probability theory and mathematical statistics, the above analysis shows the probability density function of the distribution of the number of particles [33,34]. We define d as x, with f(d) = f(x), where x denotes the particle diameter and replaces d. The integration yields [35]:

$${\displaystyle {\int }_{d_{\min }}^{d_{\max }}f\left(x\right)}dx=1-{\left(\frac{d_{\min }}{d_{\max }}\right)}^D\equiv 1$$

(19)

The above equation has the following conditions:

$${\left(\frac{d_{\min }}{d_{\max }}\right)}^D\approx 0$$

(20)

In the crushed coal rock particles studied in this paper, d_min << d_max; thus, Equation (20) holds.

Assuming that the study objects have the same density ρ and the same shape factor Kv, the percentage of particles with particle sizes less than d is:

$$\frac{M\left(<d\right)}{M_{\mathrm{t}}}=1-\frac{M\left(\ge d\right)}{M_{\mathrm{t}}}==\frac{{\left(\raisebox{1ex}{$d$}\!\left/ \!\raisebox{-1ex}{$d_{\max }$}\right.\right)}^{3-D}-{\left(\raisebox{1ex}{$d_{\min }$}\!\left/ \!\raisebox{-1ex}{$d_{\max }$}\right.\right)}^{3-D}}{1-{\left(\raisebox{1ex}{$d_{\min }$}\!\left/ \!\raisebox{-1ex}{$d_{\max }$}\right.\right)}^{3-D}}={\left(\frac{d}{d_{\max }}\right)}^{3-D}$$

(21)

where M (<d) is the mass of specimens with particle sizes less than d and M_t is the total mass of the specimen.

According to Equation (21), the following holds:

$$\overline{d}=2^{\frac{1}{D-3}}{d}_{\max }$$

(22)

Equations (17), (19), (21) and (22) can be coupled, yielding:

$$k_e=\frac{{\varphi }^3{4}^{\frac{1}{D-3}}{d}_{\max }{}^2}{36C{\left(1-\varphi \right)}^2+1.75B\left(1-\varphi \right)2^{\frac{1}{D-3}}{d}_{\max }v}$$

(23)

where B = ρ/μ. This equation represents the mathematical relationship between the fractal dimension of the particle size and the non-Darcy equivalent permeability.

4.5. Equivalent Permeability Model Test for the Crushed Coal Seepage System

The non-Darcy equivalent permeability k_e was calculated by applying the particle size fractal–percolation model developed in Section 4.4 to a pressurised fractured coal body, and the results were compared to the permeability k obtained for the pressurised fractured coal body, as shown in

Table 2. Experimental and theoretical seepage characteristics of specimens with different initial gradations.

Talbot Power Index n	Fractal Dimension D	Permeability k/m2	Equivalent Permeability ke/m2
n = 0.2	2.32	8.40 × 10−11	4.67 × 10−11
	2.42	5.48 × 10−11	3.04 × 10−11
	2.49	1.95 × 10−11	1.08 × 10−11
	2.54	1.13 × 10−11	6.09 × 10−12
	2.57	1.91 × 10−12	7.66 × 10−13
n = 0.4	2.22	3.47 × 10−11	1.93 × 10−11
	2.34	2.31 × 10−11	1.28 × 10−11
	2.43	3.34 × 10−12	1.85 × 10−12
	2.51	2.56 × 10−12	1.35 × 10−12
	2.53	2.57 × 10−13	5.57 × 10−14
n = 0.6	2.14	7.24 × 10−11	4.02 × 10−11
	2.34	4.09 × 10−11	2.27 × 10−11
	2.44	1.48 × 10−11	8.17 × 10−12
	2.50	7.14 × 10−12	3.82 × 10−12
	2.52	4.59 × 10−13	9.93 × 10−15
n = 0.8	2.06	1.47 × 10−10	8.17 × 10−11
	2.27	7.25 × 10−11	4.03 × 10−11
	2.40	3.25 × 10−11	1.79 × 10−11
	2.48	1.47 × 10−11	7.91 × 10−12
	2.51	3.91 × 10−12	1.68 × 10−12

.

The calculations show a slight difference in the two permeabilities at lower porosities. n = 0.2, n = 0.4, n = 0.6, n = 0.8 (Figure 12). When the crushed coal body has a low porosity, it is difficult for the water permeability to change as the porosity continues to decrease; thus, the permeability obtained through the test is almost constant. However, in the particle size fractal–percolation model, because the non-Darcy equivalent permeability decreases continuously with a decreasing porosity, the non-Darcy equivalent permeability k_e is slightly less than the permeability k at a low porosity. However, the permeability obtained from the tests is similar to the non-Darcy equivalent permeability calculated by applying the particle size fractal-percolation model, and the variation trend with the porosity is the same, thus verifying the practicality and accuracy of this mathematical model.

The model reveals the intrinsic relationship between the particle size distribution characteristics of the crushed coal body and the non-Darcy percolation characteristics, suggesting that the size fractal dimension is an important parameter affecting the non-Darcy percolation characteristics of the crushed coal body. Therefore, this model can be used to describe the non-Darcy percolation patterns of crushed coal bodies under pressure.

5. Conclusions

Novel experiments were used to evaluate the effects of changes in the porosity of crushed coal bodies on the permeability and structure of crushed specimens, and a mathematical model was proposed and validated.

(1)

According to the magnitude of the slope of the time-varying curve of the porosity change rate of the specimen, the change process can be divided into two stages: the rapid change stage and the slow change stage. At an axial compression of 5 mm, the porosity of a specimen with grade n = 0.2 increases from an initial value of 0.3924% to 0.5109%, an increase of 11.8%. With the extension of the permeation time, it is found that the permeation system gradually loses all fine particles that can migrate freely, which is of great significance for the study of the stability of the collapse column.

(2)

As the axial displacement increases, the axial deformation gradually increases and the porosity and permeability of the crushed specimens decrease, with the minimum porosity reaching 0.298. When the porosity decreases to approximately 0.375, crushed specimens with different initial grades have similar permeability values.

(3)

Before the sudden change in percolation, the solid particles in the crushed specimens are lost and the pore structure and permeability change; however, after the sudden change in percolation, with increasing percolation time, the skeleton and filler particles eventually stabilise and the permeability remains constant. The different specific surface area, pore distribution, pore size and the formation of an effective seepage channel of the sample under different particle size ratio also have certain influence on the seepage characteristic parameters of the collapse column structure.

(4)

At low porosity, the non-Darcy equivalent permeability ke is slightly smaller than the permeability k. However, the non-Darcy equivalent permeability that was calculated by applying the particle size fractal–percolation model is similar to the permeability obtained from the tests and has the same trend as the porosity, verifying the applicability and accuracy of this mathematical model.