The loading indenter was used to provide the axial loading. The O-shape rubber seal rings were used to close the gap. The piston was used to compress the rock specimen. The felt filtration pad was used to prevent the testing system from being polluted by the fluid, and the porous plate was used to ensure that the water flowed evenly.
In order to obtain the particle size distribution of the saturated crushed sandstone under different axial stresses, an axial force control mode was applied and the specimens were separated after the seepage test.
Figure 3 illustrates the testing procedure. Because of the movement of the overlying strata, the crushed rocks in the caved zones support different amounts of loading at different times. The resulting compression increases gradually due to the change of ground stress.
Therefore, the impact of the compression level axial stress on particle size distribution and permeability should be investigated. Therefore, the particle size distribution and permeability can be tested under six different conditions including the nonloading condition. Each set of experiments was carried out three times, and the average values of the test data were used for the analysis. The definition of a fractal can be given based on the relationship between the number and feature scale in a statistically self-similar system [ 12 , 13 ] and is given by the following equation: where is the feature scale of the rock particles, is the number of rock particles larger than , is the proportionality coefficient, and is the fractal dimension of the particle size distribution.
However, the applicability of 2 for particle size distribution analysis is limited because the accurate calculations of values are typically unavailable from conventional particle size distribution experimental data.
In order to compensate for the lack of values, Tyler and Wheatcraft [ 30 ] estimated the fractal dimension of the particle size distribution based on the following expression: where is the mass of the sandstone particles smaller than , is the total mass of the specimen, and is the maximum diameter of the sandstone particles.
From 3 , it is found that the relationship between and is linear, and the slope is. As previously suggested, if the particle size distribution of the saturated crushed sandstone in the compression test can satisfy the fractal condition, we can fit the straight line of to obtain the fractal dimension of the particle size distribution.
The Forchheimer equation [ 31 ] can be used to describe the relationship between the water pressure gradient and the flow velocity in crushed rocks [ 1 , 3 ]. For a one-dimensional non-Darcy flow, the relationship can be expressed as where is the pore water pressure gradient, is the pore water pressure, is the vertical axis going through the center of the specimen, is the kinetic viscosity of the water, is the permeability, is the water flow velocity, is the water density, and is the non-Darcy coefficient.
As shown in Figure 2 , the upstream end of the specimen is connected to the pressure intensifier tank in the MTS Such a connection could apply the required pore water pressure.
The downstream end of the specimen is connected to the atmosphere; thus the pore water pressure is equal to zero. If all parameters on the right side of 4 do not change with , then the pore water pressure gradient is a constant, which can be calculated using where is the specimen height. Therefore, 4 can be expressed as. In the above described test, we could obtain the steady water flow velocity corresponding to each required pore water pressure. Based on 6 , the permeability of the saturated crushed sandstone could be obtained by fitting the curves.
Figure 4 shows the X-ray CT images of the specimens with under different axial stresses. The particles accumulated together in a disordered way and made contact with each other in the modes of point-to-point and point-to-surface.
In addition, the pore connectivity was quite good, and there were few isolated pores. After loading was applied, many secondary particles appeared, indicating the occurrence of particle crushing. Particles were moved and rearranged, and the mode of contact was gradually transformed to surface-to-surface contact, which is relatively stable.
In particular, when the axial stress was increased, the number and size of pores decreased greatly, and the connectivity between pores became poor. Under the higher axial stress, as shown in Figure 4 f , most of the pores were compressed or filled with small particles. The residual pores were isolated, and the pore shape evolved from an unstable polygon into a stable triangle. Moreover, during the compression, larger pores were mainly distributed around larger particles, indicating that larger particles are more likely to cause larger capillary tubes for water flow.
Based on the mass percent of the rock particles in each diameter range under variable axial stresses, we can calculate the corresponding fractal dimension of the particle size distribution. Saturated crushed sandstone specimens of will be described as an example to show how the fractal dimension was calculated. First, we obtained the mass percent of the rock particles in each diameter range under different axial stresses, as shown in Table 2.
Next, from Table 2 , we calculated the values of and. Finally, according to 3 , we fit the straight lines of and calculate the fractal dimension, as shown in Figure 5.
In Figure 5 , it can be seen that the particle size distribution of the saturated crushed sandstone satisfies the fractal condition well, and all of the correlation coefficients are in the range of 0. Moreover, the fractal dimension increases monotonically with an increase in the mass percent of small particles, and there is a one-to-one correspondence between a fractal dimension value and the particle size distribution.
Thus, it can be concluded that the fractal dimension of the particle size distribution is an effective parameter to describe the particle crushing state of the saturated crushed sandstone.
Table 3 shows the calculated fractal dimension, and Figure 6 shows the fractal dimension-axial stress curves. In Figure 6 , it can be seen that the fractal dimension that ranges from 1. This is mainly due to the fact that there are a large number of large particles during the early stage of the compaction. There exist many flaws, harp corners, and the unstable contact modes between particles including point-to-point and point-to-surface , which result in the concentration of stress.
As a result, a large amount of particle crushing occurs see Figure 4 and the fractal dimension increases rapidly. In comparison, the number of large particles decreases during the later stage. The particle shapes are relatively regular, and the contacts between particles are relatively stable. Therefore, only a slight amount of particle crushing occurs and the fractal dimension increases slowly. The porosity of the saturated crushed sandstone is a measurement of the fraction of void spaces in the specimen, which can be expressed as where , , , , and are the porosity, mass, mass density, the height of the specimen during compaction, and the cross-sectional area of the cylindrical tube, respectively.
Table 4 shows the calculated porosity, and the values of those measured parameters to calculate the porosity are listed in Table 5. Figure 7 shows the porosity-fractal dimension curves. In Figure 7 , it can be seen that the porosity decreases with an increase in the fractal dimension of the particle size distribution. That is mainly due to the fact that a larger fractal dimension corresponds to a larger mass percent of small particles.
This will accelerate particle rearrangement and fill in the pores between large particles see Figure 4 , resulting in a decrease in the porosity. Moreover, the relation between the porosity and the fractal dimension can be described by a linear function: where is the porosity, is the fractal dimension of the particle size distribution, and and are the regression coefficients.
Table 6 shows the calculated permeability, and the values of parameters used to calculate the permeability are listed in Table 7. Figure 8 shows the permeability-axial stress curves. In Figure 8 , it can be seen that the permeability that ranges from 3.
The decrease process can be divided into two stages which correspond to the two stages of the fractal dimension of the particle size distribution. In the initial state, as presented in the X-ray CT results see Figure 4 , the pore size is large and the pore connectivity is quite good. As a result, the permeability is large. Many pores are compressed greatly and closed, and the pore connectivity becomes poor.
The pore size and pore connectivity change slightly, and thus the permeability decreases slowly. In addition, the pore structure becomes random and uncertain due to the particle crushing and particle rearrangement. As a result, the permeability shows some local fluctuation. The permeability is influenced by the initial gradation. Under the same axial stress, a larger Talbot power exponent corresponds to a larger permeability. This is mainly because a larger Talbot power exponent corresponds to a greater mass percent of large particles.
It is more likely to cause large capillary tubes for water flow, thus resulting in a larger permeability. Figure 9 shows the permeability-fractal dimension curves.
In Figure 9 , it can be seen that the permeability decreases with an increase in the fractal dimension of the particle size distribution, and the relation between them can be described by an exponential function where is the permeability, is the fractal dimension of the particle size distribution, and , , and are the regression coefficients.
In this study, the saturated crushed sandstone is taken as a research object. Actually, many factors such as rock properties, rock moisture content, and inner defects of rock may greatly influence the magnitude of the key axial stress. In future, a further study of the particle size distribution and permeability evolution for the rocks under different rock properties and other factors is necessary to be performed.
It is found that the fractal dimension of the particle size distribution is an effective method for describing the particle crushing state of saturated crushed sandstone under compression. However, the validity of the relationship between fractal dimension and porosity and permeability needs to be proved by theoretical analysis. Moreover, we cannot obtain a quantitative relationship between the pore structure evolution rule and the fractal dimension of the particle size distribution, which needs to be further studied.
The pore structure including the contact mode between rock particles, the number and size of pores, and the connectivity between pores changes greatly.
The authors declare that there are no conflicts of interest regarding the publication of this paper. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Article of the Year Award: Outstanding research contributions of , as selected by our Chief Editors. Read the winning articles. Journal overview. Special Issues. Academic Editor: Qinghui Jiang. Received 20 Jun Revised 20 Oct Accepted 21 Nov Published 10 Jan Introduction Due to the large permeability of crushed rocks, flow catastrophes and water inrush accidents can be easily triggered in the discontinuous zones of aquifers in underground coal mines [ 1 , 2 ].
Experimental Materials and Testing Methods 2. Figure 1. Specimen number Talbot exponent Mass in each diameter range g 2. Table 1.
The details of the mass amount of the sandstone particles in each diameter range. Figure 2. Testing system. Note: A: pressure sensor, B: relief valve, C: drainage, D: regulator, E: pressure difference sensor, F: load controller, S1 to S15 are switches, loading indenter, O-shape rubber seal rings, piston, felt filtration pad, porous plate, cylindrical tube, rock specimen, and base plate.
Figure 3. Figure 4. The formed process of isolated pores in the specimen under the increased axial stresses. Table 2. The mass percent of the rock particles in each diameter range of the saturated crushed sandstone with under variable axial stresses.
Figure 5. Fitting process curves of the fractal dimension with. Table 3. Fractal dimension of the particle size distribution under variable axial stresses. Figure 6. The increase process of the fractal dimension with the axial stress.
Table 4. Table 5. The values of those measured parameters to calculate porosity. Figure 7. To obtain the best experience, we recommend you use a more up to date browser or turn off compatibility mode in Internet Explorer. In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript. MANY attempts have been made to find a relation between permeability and other measurable properties of porous materials such as particle-size and porosity.
The Kozeny equation, which depends on surface area of particles and porosity, has come into rather general use in fields concerned with flow of water, oil and gases and a considerable literature has developed around it 1.
Its weaknesses are that it is unsuited to material having a wide range of pore-sizes, and that it makes use of an empirical factor which departs seriously from its accepted value in consolidated materials.
Attention has also been given to the possibility of using pore-size instead of particle-properties, and Childs and Collis George 2 have developed a permeability equation on this basis. An empirical factor is involved and this has not been widely tested for constancy in different materials. Carman, P. Childs, E. Purcell, W. Google Scholar. Wyllie, M. CAS Google Scholar. Day, P. Soil Sci. Moore, R. Article Google Scholar. Baver, L. Download references.
You can also search for this author in PubMed Google Scholar. Reprints and Permissions. Permeability and the Size Distribution of Pores. Nature , — Download citation. Issue Date : 28 September Anyone you share the following link with will be able to read this content:.
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