# Experimental Study on Sensitivity of Porosity to Pressure and Particle Size in Loose Coal Media

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials

_{1}of the Guobei Coal Mine in the Huaibei mining area. The altitude of working face 8105-1 is between −683.6 m and −890.0 m, and the corresponding ground surface altitude varies from 29.6 m to 31.5 m. The structure of coal seam 8

_{1}is complex, as shown in Figure 1.

#### 2.2. Methods

#### 2.2.1. Porosity in the Particle Matrix

_{p}is the particle matrix porosity; V

_{1}is the pore volume (mercury volume intruded at the maximum experimental pressure attained, 414 MPa) and V

_{2}is the sample volume which can be derived by Equation (2)

_{Hg}is the density of mercury at a certain temperature; w

_{1}is the mass of empty cell; w

_{2}is the mass of empty cell and sample; w

_{3}is the mass of mercury, empty cell and sample and after filling with mercury under a certain pressure; w

_{4}is the mass of mercury and empty cell after empty cell filled with mercury.

_{2}gas isotherm adsorption measurements were carried out at 77 K (−196 °C). The pore volume and pore size distribution can be obtained from the low temperature nitrogen adsorption experiments.

#### 2.2.2. Total Porosity of Loose Coal

_{t}is the total porosity of coal or rock (%); ρ

_{b}is the bulk density of coal or rock, i.e., the ratio of mass to total volume of coal or rock (including the volume of skeleton and all pore spaces, g/cm

^{3}) and ρ is the true density or mineral particle skeletal density of coal or rock (the ratio of mass to skeleton volume, not including the pore spaces, g/cm

^{3}).

_{b}) of coal samples at different pressures, the UTM5504 universal testing machine was used. The experimental equipment used and a schematic diagram of the method were shown in Figure 6.

- (1)
- Forward loading stage: A loading speed of 2 mm/min was used, and after the specified pressure (Determined according to the in situ stresses—5, 10, 15, 20, and 25 MPa—was selected in this test because the in situ stress of Guobei Coal Mine 8105-1 working face is 20 MPa) level was reached, the force applied was kept constant for 30 min. The loading time was referenced by some scholars’ briquette compressing schemes [24,25,36,37]. The forward decreased displacement of indenter h
_{1}was automatically recorded by the testing machine. At each initial experiment, the coal sample should be filled with the mold, that is, the initial position of the bottom surface of the indenter was flush with the top of the coal briquette mold, as shown in Figure 6d. The purpose of the operation is to conveniently calculate the height of the coal sample after compressing. After this stage, h_{coal}= h − h_{1}. h is the height of the briquette mold excluding the base embedded in the mold. - (2)
- Forward unloading rebound stage: The unloading speeds were set to 15 N/s corresponding to experimental pressures of 5 MPa, 30 N/s→10 MPa, 45 N/s→15 MPa, 60 N/s→20 MPa, and 75 N/s→25 MPa. The purpose is to keep the consistent unloading time at each pressure. When the force returned to zero, the compressed coal sample rebounded and the forward rebound amount of the coal sample h
_{2}was recorded. - (3)
- Reverse loading stage: The briquette mold was reversed, and the pressure was applied continuously. The loading speed was set to 2 mm/min and after the specified pressure level (5, 10, 15, 20, and 25 MPa) was reached the applied force was kept constant for 30 min. The reverse loading parameters were exactly the same as the parameters in the forward loading stage. The reverse decreased displacement of indenter h
_{3}was recorded. - (4)
- Reverse unloading rebound stage: The unloading speeds were set to 15 N/s corresponding to experimental pressures of 5 MPa, 30 N/s→10 MPa, 45 N/s→15 MPa, 60 N/s→20 MPa, and 75 N/s→25 MPa. When the reverse force returned to zero, the compressed coal sample rebounded and the reverse rebound amount h
_{4}was recorded. - (5)
- The mold was again turned upside down and the next level of pressure was applied. The previous four stages were repeated to complete the next pressure level tests.

_{b}is the bulk density of coal under a certain overburden pressure (g/cm

^{3}); m is the mass in grams of the coal sample in the briquette mold (the coal sample filled the mold at the beginning of the test); and d is the inner diameter of the briquette mold (cm). The inner diameter used in this experiment was 4.960 cm.

_{1}is the forward decreased displacement (cm) of the indenter under a certain overburden pressure; h

_{2}is the forward rebound amount (cm) of the coal sample when the overburden pressure returns to zero; and h

_{3}is the reverse decreased displacement (cm) of the indenter under a certain overburden pressure. The result of h − (h

_{1}+ h

_{3}− h

_{2}) is the height of the compressed coal samples under each pressure.

_{4}of the previous pressure level. For example, in the experiment scheme of this article, when the pressure is 5 MPa, the bulk density can be calculated directly using Equation (4), but when 10 MPa, the reverse rebound amount h

_{4}of 5 MPa needs to be subtracted when calculating the height of the compressed coal sample. Because the 10 MPa forward loading stage was followed by 5 MPa reverse unloading rebound stage.

## 3. Results

#### 3.1. Characterization of Total Porosity of Loose Coal

#### 3.1.1. True Density

^{3}, respectively. The average true density of minerals in coal is approximately 3.00 g/cm

^{3}, and the higher the inorganic mineral content, the higher the true density of the coal [38]. The associated mineral compositions of the coal samples were analyzed by the D8 ADVANCE X-ray diffractometer (Bruker AG, Karlsruhe, Germany), as shown in Figure 7.

#### 3.1.2. Porosity in the Particle Matrix

_{1}determined from the combined test of MIP and low temperature nitrogen adsorption was 4.23%. With 50 nm as the demarcation point of MIP and low temperature nitrogen adsorption [39,40], volume data for pores (diameter > 50 nm) used MIP data, and pores (diameter < 50 nm) used low temperature nitrogen adsorption data. The comparison between combined methods and separate MIP was shown in Figure 8a. In the figure, in order to show the comparison and correction more intuitively, the pore diameter data of low temperature nitrogen adsorption was converted into mercury intrusion pressure. Cumulative porosity and porosity distribution curves after correction were shown in Figure 8b. The matrix porosity of this sample is mainly distributed as transitional pores (reference to Hodot pore grading [41]), with pore throat diameters in the range of a 20–40 nm, and 90–347 μm diameter fractures and pores (visible to naked eye) which have an order of magnitude size difference compared to the transitional pores are also evident. There are also few abrupt peaks on the porosity distribution curve which may be due to the randomness of the samples.

#### 3.1.3. Total Porosity of Loose Coal

^{3}was used for γ and the in situ stress at the sampling site was ≈20 MPa. The experimental results indicate a total porosity value of 10.22% for the sample from the 8

_{1}coal seam at 20 MPa.

_{1}coal seam was approximately 4.23%. The remaining porosity of 5.99% for samples subjected to 20 MPa of overburden pressure was obtained from the inter-particle voids and the smaller pores (diameter < 1.77 nm) not measurable with the low temperature nitrogen adsorption experiment. In this article, the single coal sample of 5–15 mm was used in the MIP experiment, so the porosity result from MIP experiment does not include inter-particle voids. It can also be seen from the mercury intrusion curve that there is no obvious inflection point at low pressure.

#### 3.2. Effects of Pressure and Particle Size on Total Porosity

#### 3.2.1. Pressure Effects on Total Porosity

_{t}is the total porosity of loose coal at a given pressure (%), σ is the pressure (MPa), the sum of a and c represents the total porosity of loose coal when the pressure is 0 MPa (%), and b represents the compression coefficient (MPa

^{−1}).

#### 3.2.2. Particle Size Effects on Total Porosity

_{50}was selected as the average particle size of the segment. This is the particle size corresponding to 50% of the cumulative PSD in each segment of the three size classes. The fitting curve in Figure 4 shows that the D

_{50}results for each segment are 0.54, 3.15, and 8.57 mm, respectively. From these data, it was possible to construct curves of total porosity change vs. initial particle size for each pressure interval, as shown in the line graph in Figure 11. The coal samples after each stage of loading screening was performed. Using the obtained particle size after loading, the curves of total porosity change vs. the particle size of the compacted samples were plotted, as shown in the histogram in Figure 11.

_{t}represents the total porosity; d represents the initial particle size and A and B are fitting constants.

#### 3.2.3. Sensitivity of Total Porosity to Pressure and Particle Size

_{t}), as expressed in Equation (7). The greater the value of change in Δn

_{t}as some other factor changes, the higher the sensitivity of total porosity to that factor, as

_{0}is the total porosity of coal samples in the 5–15 mm initial particle size range measured at 0 MPa pressure and n

_{ij}is the total porosity measured for any particle size and pressure.

## 4. Discussions

#### 4.1. Comparison with the Method of Compressing Briquette and Testing Porosity

_{4}), but the method proposed in this paper eliminates the impact of the reverse rebound amount, not counting.

#### 4.2. Quality of Measurement Results

_{i}, as shown in Equation (8)

_{c}(n

_{t}) is the combined standard uncertainty of output estimate n

_{t}; ∂f/∂x

_{i}is the partial derivative (sensitive coefficient) with respect to input quantities X

_{i}(ρ, m, d, h, h

_{1}, h

_{3}, h

_{2}) of functional relationship f between measurand n

_{t}and input quantities X

_{i}on which n

_{t}depends, u(x

_{i}) is the standard uncertainty of input estimated x

_{i}that estimates input quantity X

_{i}, u(s) is the standard uncertainty of the measurement repeatability of output estimate n

_{t}, equal to the experimental standard deviation of the mean n

_{t}.

_{c}can be universally used to express the uncertainty of a measurement result, it only corresponds to the standard deviation, and the measurement result y ± u

_{c}represented by it has a low level of confidence. The additional measure of uncertainty that meets the requirement of providing an interval of the high level of confidence is termed expanded uncertainty and is denoted by U. The expanded uncertainty U is obtained by multiplying the combined standard uncertainty u

_{c}(y) by a coverage factor k: U = ku

_{c}(y).

_{c}(n

_{t}) = 2 × 0.17% = 0.34%.

_{t}= 10.22% ± 0.34%, p = 95%

_{c}(n

_{t}). The main sources of u(s) are the randomness of the coal samples, and the measurement repeatability of the coal sample mass. The value of this part of the uncertainty can be reduced by increasing the number of repetitions of the measurements.

#### 4.3. Engineering Significance, Novelty, Applicability, and Scalability of the Method

## 5. Conclusions

- (1)
- A new method for characterizing total porosity in loose media subjected to overburden pressure is proposed. It is based on the functional relationship between total porosity, true density, and bulk density.
- (2)
- After testing, the porosity of loose coal from the Guobei Coal Mine at 20 MPa in situ stress is found to be ≈ 10.22%. The total porosity experiences a downward trend as pressure increases for a fixed particle size, and the total porosity and pressure obey an attenuated exponential function. The decrease in total porosity with initial single particle sizes (0–2, 2–5, and 5–15 mm) is similar to that with increasing pressure, with steep curves of total porosity vs. pressure evident. There is reduction in the rate of total porosity decrease with increasing pressure with a mixed particle sizes.
- (3)
- At each selected pressure, the total porosity increases with increasing initial particle size (large initial particle size correspond to low degree of on-site coal fragmentation), and the total porosity and initial particle size obey a power function. The rate of total porosity increase becomes gradually reduced as particle size increases at higher stress levels. The curve of initial particle size vs. total porosity approximates a horizontal line when the pressure exceeds 20 MPa, and can thus be considered indicative of total porosity being insensitive to changes in initial particle size or the degree of on-site coal fragmentation.
- (4)
- When pressures are low (e.g., burial conditions are shallow), it is found that total porosity is greatly reduced and is highly sensitive to the increase in pressure. However, total porosity is less sensitive to pressure at higher stress levels (e.g., burial conditions are deep). The effect of particle size on the total porosity reduction rate in the loose coal is not significant irrespective of the pressure conditions (e.g., low or high). In general, the sensitivity of the total porosity to pressure is found to be significantly higher than sensitivity to particle size.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 3.**(

**a**) On-site sampling photographs and (

**b**–

**d**) surface micromorphology of 5–15 mm, 2–5 mm, 0–2 mm coal particles from the 8

_{1}coal seam.

**Figure 6.**(

**a**) Forward loading photograph; (

**b**) reverse loading photograph; (

**c**) mold schematic diagram; (

**d**) schematic diagram of initial experimental coal samples; and (

**e**) schematic diagram of the test apparatus and methodology for determining bulk density of coal at different pressures.

**Figure 8.**Results of pore space characterization using MIP and low temperature nitrogen adsorption: (

**a**) pore volume and (

**b**) porosity in the coal matrix.

**Figure 10.**Curves of total porosity change with increasing pressure for specified particle size classes.

**Figure 11.**Curves of total porosity change vs. initial particle size and the particle size of the compacted samples.

**Figure 12.**(

**a**) Time–displacement curves determined during loading and unloading; and (

**b**) comparison of porosity differences between the method of compressing briquette and then testing porosity and the method proposed in this paper.

Mesh Diameter (mm) | Grader Retained Mass m_{i} (g) | Grader Retained Percentage (m_{i}/2000) (%) | Accumulated Retained Percentage (%) |
---|---|---|---|

31.5 | 0 | 0 | 0 |

16 | 124.77 | 6.2385 | 6.2385 |

9.5 | 131.25 | 6.5625 | 12.801 |

4.75 | 235.18 | 11.759 | 24.56 |

2.36 | 328.3 | 16.415 | 40.975 |

1.18 | 262.71 | 13.1355 | 54.1105 |

0.6 | 335.51 | 16.7755 | 70.886 |

0.3 | 189.61 | 9.4805 | 80.3665 |

0.15 | 183.05 | 9.1525 | 89.519 |

0.088 | 173.26 | 8.663 | 98.182 |

<0.088 | 15.98 | 0.799 | 98.981 |

Minerals | Quartz | Kaolinite | Others |
---|---|---|---|

Weight percentage | 12.04% | 9.47% | 3.79% |

Size | 0–2 mm | 2–5 mm | 5–15 mm | Raw Coal |
---|---|---|---|---|

True density | 1.562 g/cm^{3} | 1.632 g/cm^{3} | 1.827 g/cm^{3} | 1.636 g/cm^{3} |

Size | Fitting Formula | R-Square |
---|---|---|

0–2 mm | ${n}_{t}=0.098+0.351{e}^{-0.217\sigma}$ | $0.97848$ |

2–5 mm | ${n}_{t}=0.095+0.371{e}^{-0.199\sigma}$ | $0.98336$ |

5–15 mm | ${n}_{t}=0.099+0.400{e}^{-0.211\sigma}$ | $0.98217$ |

Raw coal | ${n}_{t}=0.091+0.255{e}^{-0.162\sigma}$ | $0.98215$ |

Size | 5–15 mm | 2–5 mm | 0–2 mm |
---|---|---|---|

Pressure | |||

0 MPa | 0 | 6.54% | 10.03% |

5 MPa | 55.96% | 56.78% | 60.18% |

10 MPa | 66.97% | 67.96% | 68.76% |

15 MPa | 73.64% | 74.19% | 75.04% |

20 MPa | 79.41% | 79.55% | 79.73% |

25 MPa | 83.74% | 84.06% | 84.06% |

Standard Uncertainty Component u(x_{i}) | Source of Uncertainty | Value of Standard Uncertainty u(x_{i}) | Sensitive Coefficient: c_{i} = ∂f/∂x_{i} | Component of u_{c}(n_{t}): u_{i}(n_{t}) = |c_{i}|u(x_{i}) (%) |
---|---|---|---|---|

u(ρ) | Instrument uncertainty (±0.03% of the indication) | 2.83 × 10^{−4} g/cm^{3} | 0.14654 | 0.00438 |

Measurement repeatability (±0.01% of the indication) | 4.15 × 10^{−5} g/cm^{3} | |||

u(m) | Instrument uncertainty (±0.02 g) | 0.0115 g | 0.00111 | 0.00128 |

u(d) | Instrument uncertainty (±0.002 cm) | 0.00115 cm | 0.09667 | 0.0869 |

Measurement repeatability | 0.00892 cm | |||

u(h) | Instrument uncertainty (±0.002 cm) | 0.00115 cm | 0.00314 | 0.00286 |

Measurement repeatability | 0.00904 cm | |||

u(h_{1}) | Instrument uncertainty (±0.5% of the indication) | 0.00730 cm | 0.00314 | 0.00229 |

u(h_{3}) | 0.00154 cm | 0.000484 | ||

u(h_{2}) | 0.000427 cm | 0.000134 | ||

u(s) | - | - | - | 0.140 |

_{c}

^{2}(n

_{t}) = ∑u

_{i}

^{2}(n

_{t}) + u

^{2}(s) = 2.74 × 10

^{−5}; u

_{c}(n

_{t}) = 0.17%.

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## Share and Cite

**MDPI and ACS Style**

Zhang, C.; Zhang, N.; Pan, D.; Qian, D.; An, Y.; Yuan, Y.; Xiang, Z.; Wang, Y.
Experimental Study on Sensitivity of Porosity to Pressure and Particle Size in Loose Coal Media. *Energies* **2018**, *11*, 2274.
https://doi.org/10.3390/en11092274

**AMA Style**

Zhang C, Zhang N, Pan D, Qian D, An Y, Yuan Y, Xiang Z, Wang Y.
Experimental Study on Sensitivity of Porosity to Pressure and Particle Size in Loose Coal Media. *Energies*. 2018; 11(9):2274.
https://doi.org/10.3390/en11092274

**Chicago/Turabian Style**

Zhang, Chenghao, Nong Zhang, Dongjiang Pan, Deyu Qian, Yanpei An, Yuxin Yuan, Zhe Xiang, and Yang Wang.
2018. "Experimental Study on Sensitivity of Porosity to Pressure and Particle Size in Loose Coal Media" *Energies* 11, no. 9: 2274.
https://doi.org/10.3390/en11092274