# Size Effects of Rough Fracture Seepage in Rocks of Different Scales

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Seepage Experiments with Different Fracture Lengths and Fracture Inclinations

#### 2.1. Experimental Equipment

- (1)
- Coal-sample compressive strength experiments. To ensure that the coal sample is subjected to a variety of stresses in the seepage experiment, and to reduce the interference of the anisotropy of the coal sample in the experiment, the seepage loading scheme should be set to ensure that the maximum loading stress is within the elastic range of the coal sample as far as possible. To prevent the coal sample from being damaged during the loading process and changing its seepage path, a compressive strength experiment on the cylindrical coal sample should be conducted first.
- (2)
- Seepage experiments with different fracture lengths. Because of the need to investigate the effect of fracture length and the relationship between fracture length and other factors when coupled with each other, in this part of the experiment, the different lengths of fractures that exist within the coal or rock in a realistic environment were simulated by processing multiple sets of prefabricated fractures of different lengths.
- (3)
- Seepage experiments with different fracture inclination angles. To consider the influence of multiple factors, the fracture geometry, surrounding pressure, axial pressure, and fracture inclination were coupled; therefore, seepage experiments with different fracture inclination angles were required.

_{1,max}= 150 MPa, radial load σ

_{2,max}= σ

_{3,max}= 25 MPa, and pressure upstream of the liquid seepage in the seepage system p

_{w,max}= 20 MPa. The equipment can conduct triaxial rock mechanics experiments under different stress paths, seepage experiments, and data acquisition using high-precision sensors and other equipment. The seepage flow experimental equipment is shown in Figure 2, and a schematic of the equipment is shown in Figure 3.

#### 2.2. Sample Preparation

#### 2.3. Experimental Programme

_{1}, surrounding pressures σ

_{2}= σ

_{3}, seepage pressure p

_{w}, and internal permeation pressure p

_{w}

_{0}acting on the fracture surface after water infiltration into the prefabricated fractures through the hydraulic conductivity holes [21], as shown in Figure 6.

_{1}and radial stress σ

_{2}and σ

_{3}at a rate of 1 MPa/min, followed by displacement-controlled loading with axial stress σ

_{1}applied at a rate of 0.01 mm/min and simultaneous application of radial stress σ

_{2}and σ

_{3}to achieve experimental standard loading conditions. When axial stress σ

_{1}and radial stress σ

_{2}and σ

_{3}are stable, the percolation pressure pw is applied via a level flow pump using a constant flow mode of 5 mL/min controlled by pressure. After all the stresses have stabilized, the seepage experimental data collection is started. The stress paths are shown in Figure 6.

_{1}and b

_{2}, respectively. In the cubic law, the flow through the cut or fracture can be expressed as

^{3}/s), b is the cut width (b

_{1}) or fracture width (b

_{2}) (m), s is the width of seepage through the matrix (m), and $\u2206p$ is the pressure difference between upstream and downstream seepage (Pa).

#### 2.4. Compressive Strength Test of Coal

_{c}of the coal sample selected for this experiment was calculated to be approximately 39.6 MPa. The slope of the curve within the elastic phase reveals that the coal sample is stiffer and less plastic, with more pronounced brittleness [21]. In combination with the above experimental results, the stress loading scheme can be determined, as shown in Table 2.

#### 2.5. Seepage Law of Coal Fracture under Size Effect

_{1}= 6 MPa, the confining pressure σ

_{2}= σ

_{3}= 4 MPa curve is considered as an example. When the crack length is 10 mm, the coal sample permeability is 4.04 × 10

^{−18}m

^{2}, and with an increase in fracture length, when the fracture length is 70 mm, the coal sample permeability increases to 1.15 × 10

^{−17}m

^{2}. Under the same stress condition, the coal sample permeability increases with an increase in the fracture length in the coal sample fracture seepage. A planar seepage model was developed using COMSOL software. Add a line segment to the planar model and define it as a fracture. Simulate the seepage of coal by setting parameters and boundary conditions. Water flows in from one end of the planar model and out from the other under a set pressure. The analysis of the seepage velocity of the seepage field leads to the following conclusions. The theoretical analysis and numerical simulation results show that when there are many or large fractures in the coal sample the seepage path will change, and a small part of the water will flow through the pores of the coal sample matrix, but mainly through the fractures, as shown in Figure 10. The existence of fractures can be understood through change in the seepage path length. The pressure gradient of seepage should change accordingly, and the permeability should show a linear change in theory; however, the permeability change results show nonlinear changes in actual experiments. After further analysis, the reason for this phenomenon is that, under the action of axial stress and confining pressure, the internal micropores of the coal body are compressed, resulting in the pores becoming smaller. The water absorbed in the coal body is squeezed and diffused into the fractures and becomes free water, which moves with the fractures under seepage pressure. While the applied stress produces new microcracks that expand and extend, the more obvious role is to close the more obvious cracks. This phenomenon is particularly evident when the crack size is large. This can explain the nonlinear change in coal sample permeability with the increase in fracture length, and also proves that there is a size effect of coal seepage. The seepage results of coal samples having the same fracture length were analysed. When the fracture length was 10 mm, the variation range of coal sample permeability under different stress conditions was 2.62 × 10

^{−18}~4.04 × 10

^{−18}m

^{2}, variation 1.42 × 10

^{−18}m

^{2}; when the crack length was 70 mm, the variation range of the coal sample permeability under different stress conditions was 7.35 × 10

^{−18}~1.11 × 10

^{−17}m

^{2}, variation 3.75 × 10

^{−18}m

^{2}. In the experimental group with a small fracture length, the change in permeability caused by the change in stress condition was small; in the same experimental group with a large fracture length, the change in permeability caused by the change in stress conditions increased significantly.

#### 2.5.1. Seepage of Coal Samples with Multiple Fracture Lengths under Different Axial Pressures

_{2}= σ

_{3}= 4 MPa, the axial pressure σ

_{1}changes to 6, 8, and 10 MPa. With the increase in fracture length, the permeability of the coal samples changes by 4.04 × 10

^{−18}to 1.11 × 10

^{−17}m

^{2}, 3.38 × 10

^{−18}~9.57 × 10

^{−18}m

^{2}, 3.36 × 10

^{−18}~9.51 × 10

^{−18}m

^{2}, respectively, and the permeability change is approximately 1.8 times; When confining pressure σ

_{2}= σ

_{3}= 5 MPa, the axial pressure σ

_{1}changes to 6 MPa, 8 MPa, and 10 MPa. With the increase in fracture length, the permeability of the coal sample changes from 3.11 × 10

^{−18}to 8.39 × 10

^{−18}m

^{2}, 2.99 × 10

^{−18}~8.11 × 10

^{−18}m

^{2}, 2.89 × 10

^{−18}~7.97 × 10

^{−18}m

^{2}, respectively, and the permeability change is approximately 1.7 times. When the confining pressure σ

_{2}= σ

_{3}= 6 MPa, the axial pressure σ

_{1}changes to 6, 8, and 10 MPa. With the increase in fracture length, the permeability of the coal samples changed from 2.81 × 10

^{−18}to 7.62 × 10

^{−18}m

^{2}, 2.70 × 10

^{−18}~7.42 × 10

^{−18}m

^{2}, 2.62 × 10

^{−18}~7.42 × 10

^{−18}m

^{2}, respectively, and the permeability change was approximately 1.7 times. Further analysis leads to the following conclusions: changing the axial pressure σ

_{1}when the surrounding pressures σ

_{2}and σ

_{3}are small, the overall change in permeability is more obvious; when the surrounding pressures σ

_{2}and σ

_{3}are large, changing the axial pressure σ

_{1}weakens its effect on permeability. Under the same stress conditions, the seepage size effect of the coal samples is more obvious.

#### 2.5.2. Seepage of Coal Samples with Multiple Fracture Lengths under Different Confining Pressures

_{1}is 6 MPa and the surrounding pressures σ

_{2}and σ

_{3}are 4 MPa, 5 MPa, and 6 MPa, respectively, the permeability variations in the experimental group with fracture lengths of 10 mm and 70 mm range from 2.81 × 10

^{−18}to 4.04 × 10

^{−18}m

^{2}and 7.62 × 10

^{−18}to 1.11 × 10

^{−17}m

^{2}, respectively, showing a permeability variation of approximately 2.83 times. When the axial pressure σ

_{1}is 8 MPa and the surrounding pressures σ

_{2}and σ

_{3}are 4 MPa, 5 MPa, and 6 MPa, respectively, the permeability variations within the experimental group of 10 mm and 70 mm fracture lengths range from 2.70 × 10

^{−18}to 3.38 × 10

^{−18}m

^{2}and 7.42 × 10

^{−18}to 9.57 × 10

^{−18}m

^{2}, respectively, and the permeability variation is approximately 3.2 times. When the axial pressure σ

_{1}is 10 MPa and the surrounding pressures σ

_{2}and σ

_{3}are 4 MPa, 5 MPa, and 6 MPa, respectively, the permeability variation ranges from 2.62 × 10

^{−18}to 3.36 × 10

^{−18}m

^{2}and 7.35 × 10

^{−18}to 9.51 × 10

^{−18}m

^{2}within the experimental group for fracture lengths of 10 mm and 70 mm, respectively. The permeability variation is approximately 2.9 times. Comparing the dimensional effect of axial stress with the dimensional effect of the envelope pressure, the change in envelope pressure is seen to have a more pronounced effect on the dimensional effect of permeability. When the surrounding pressure increases, the fractures within the coal are squeezed and the fracture width is further reduced, resulting in a more pronounced change in permeability when other conditions remain constant. From the above analysis, a size effect is seen in the seepage of the coal samples and the surrounding pressure σ

_{2}and σ

_{3}has a significant effect on the permeability and exacerbates the size effect of the seepage to a certain extent.

#### 2.6. Fracture Seepage Patterns in Coal at Different Fracture Dips

_{1}= 6 MPa and an enclosing pressure σ

_{2}= σ

_{3}= 4 MPa as an example, the permeability of the coal sample was reduced from 1.3 × 10

^{−17}m

^{2}to 4.8 × 10

^{−18}m

^{2}, a reduction of approximately 63%. Under the same inclination angle, the permeability of the coal samples was larger when the axial pressures σ

_{1}, σ

_{2}, and σ

_{3}were smaller. For example, when the inclination angle was 0°, the permeability of the coal sample varied from 9.1 × 10

^{−18}to 1.3 × 10

^{−17}m

^{2}under different stress conditions with a larger range of variation. When the inclination angle was 90°, the permeability of the coal sample varied from 2.6 × 10

^{−18}to 4.8 × 10

^{−18}under different stress conditions, which is a smaller range of variation. When the angle between the fractures and the laminae is small, the effect of different stress conditions on the seepage of coal samples is more pronounced, and the permeability is greater, which can also explain why the majority of seepage of coal samples occurs in the fractures along the laminae of coal samples.

## 3. Numerical Simulation Study of the Fracture Seepage Field under Size Effects

#### 3.1. Fracture Seepage Simulation

#### 3.1.1. Seepage Phenomena under Size Effects

#### 3.1.2. Seepage Patterns for Different Fracture Widths under Size Effects

^{−3}m

^{3}s

^{−1}and 10

^{−2}m

^{3}s

^{−1}orders of magnitude, and the overall permeability is between 10

^{−9}m

^{3}s

^{−1}and 10

^{−8}m

^{3}s

^{−1}orders of magnitude, and this result has a large order of magnitude difference with the presence of non-through fracture seepage flow in coal samples. This indicates that in the presence of through-fractures, seepage occurs mainly within the through fractures. When the fracture width is between 1 and 2 mm and the fracture roughness is 5.5, the permeability variation of the fracture at small lengths ranges from 1.5 × 10

^{−8}m

^{2}to 6.0 × 10

^{−8}m

^{2}. When the fracture width is between 1 and 2 mm and fracture roughness is 15.5, the permeability variation of the fracture at small lengths ranges from 5.3 × 10

^{−9}m

^{2}to 2.1 × 10

^{−8}m

^{2}, which shows that the fracture width has a greater influence on the fracture seepage. When the fracture roughness was kept constant and the fracture width increased, the change in the rock fracture permeability became increasingly obvious.

#### 3.1.3. Fracture Seepage for Different Fracture Roughness Values with Size Effect

^{−8}m

^{2}to 6.024 × 10

^{−8}m

^{2}at small lengths and 5.919 × 10

^{−8}m

^{2}to 5.923 × 10

^{−8}m

^{2}at large lengths. When the fracture roughness is 15.5 (roughness is larger), the fracture permeability varies from 2.076 × 10

^{−9}m

^{2}to 2.137 × 10

^{−8}m

^{2}at small lengths and from 2.098 × 10

^{−8}m

^{2}to 2.102 × 10

^{−8}m

^{2}at large lengths. This shows that there is a size effect on fracture seepage at different fracture roughnesses.

## 4. Fracture Geometry Stress Multifactor Analysis

#### 4.1. Analysis Method

_{1}, x

_{2}, …, x

_{p}are the independent variables and β are the coefficients; $\epsilon $ represents random errors and satisfies the following conditions: obeying a normal distribution, the assumption of unbiasedness (expectation of zero), equal variance of the random error variables, and the random error variables are independent of each other.

#### 4.2. Results of the Multi-Factor Coupling Analysis

^{2}for the regression analysis of the laboratory experimental results is 0.628. This indicates that the length of small fractures, surrounding pressure and fracture dip express up to 62.8% of the permeability. This indicates that their correlation is high. The Durbin–Watson test result is 1.931, indicating good independence between the fracture length, surrounding pressure, and fracture dip (the Durbin–Watson test result is between 0 and 4, and the data independence meets the requirement) [30,31,32]. They can be used independently as influencing factors affecting permeability. In the numerical simulation data, the adjusted R

^{2}value for this regression analysis was 0.868, indicating that the independent variables of length of large fractures, fracture width, and fracture roughness explained up to 86.8% of the dependent variable permeability. This correlation is relatively high. The Durbin–Watson test result was 1.198, indicating that the length of the large fractures, fracture width, and fracture roughness were independent of each other [33].

^{2}), ${x}_{1}$ is fracture length (mm), ${x}_{2}$ is the surrounding pressure (MPa), ${x}_{3}$ is the fracture inclination (°), ${y}_{2}$ is the permeability of fractures in large-length coal mass (m

^{2}), ${x}_{4}$ is the fracture length (m), ${x}_{5}$ is the fracture roughness, and ${x}_{6}$ is the fracture width (mm).

## 5. Main Conclusions and Recommendations

- (1)
- There was a significant size effect on coal seepage. In small-size fracture seepage from 10 to 70 mm, permeability increases with increasing fracture length, with an overall increase of approximately 1.8 times, with a trend of positive correlation and non-linear variation, which gradually stabilizes. In large fracture seepage from 1 to 30 m, permeability decreases and then increases with increasing fracture length, with an overall variation of approximately 0.03 times, which is non-linear and gradually stabilizes. With the above conclusions, it can be found that there is a significant size effect in the fracture seepage of coal or rock.
- (2)
- The overall change in permeability for different fracture lengths in large-size fracture seepage from 1 to 30 m is divided into three stages: the significant change stage, stabilization stage, and stability stage. These stages were within the ranges of 0–8 m, 8–23 m, and 23–30 m, respectively. There is a certain critical size value, and, when reaching this size, the permeability reaches a steady state. All other things being equal, the critical size of the seepage increases when fracture width increases and decreases when fracture roughness increases.
- (3)
- Permeability decreases with increasing stress under the action of perimeter and axial pressures. When the amount of change in circumferential and axial pressure is the same, the change in permeability due to circumferential pressure is approximately 3.5 times greater than that due to the axial pressure. For fractures, stresses perpendicular to the fracture direction have a greater effect on fracture permeability.
- (4)
- Regression analysis was carried out for some of the influencing factors of seepage from small-sized fractures and some of the influencing factors of seepage from large-sized fractures, respectively. The following conclusions were obtained. The sensitivity ratio of fracture seepage to fracture length, surrounding pressure and fracture dip was 0.93:0.89:1 for the small-size fractures. The influences of these factors were similar. In large-size fractures, the sensitivity factor ratio of fracture seepage to fracture length, fracture roughness, and fracture width is 0.0014:0.83:1. Fracture length has less influence on fracture seepage in large sizes.
- (5)
- There are still a number of unresolved issues regarding the study of fracture seepage in coal [34,35]. The next step is to consider research in terms of structural changes and stress changes in the coal and explore the dynamic changes in permeability at various stages of the seepage process under the influence of anisotropy and other factors. The effect of cyclic loading and unloading on seepage in different fracture geometries can also be investigated in subsequent studies to further explore the fracture seepage characteristics.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 9.**Change of coal sample seepage characteristics with fracture length under different stress conditions.

**Figure 11.**Size effect of coal sample seepage under different axial pressures. (

**a**) σ

_{2}= σ

_{3}= 4 MPa. (

**b**) σ

_{2}= σ

_{3}= 5 MPa. (

**c**) σ

_{2}= σ

_{3}= 6 MPa.

**Figure 12.**Effect of seepage sizes of coal samples under different surrounding pressure conditions. (

**a**) σ

_{1}= 6 MPa. (

**b**) σ

_{1}= 8 MPa. (

**c**) σ

_{1}= 10 MPa.

**Figure 17.**Coal fracture permeability versus fracture length and coal fracture permeability versus fracture length at different widths. (

**a**,

**b**) Flow and permeability for different fracture widths at JRC = 5.5. (

**c**,

**d**) Flow and permeability for different fracture widths at JRC = 10.5. (

**e**,

**f**) Flow and permeability for different fracture widths at JRC = 15.5.

**Figure 18.**Trends in permeability under different seepage conditions. (

**a**) Stages of change in permeability for different fracture widths at JRC = 5.5. (

**b**) Stages of change in permeability for different fracture widths at JRC = 10.5. (

**c**) Stages of change in permeability for different fracture widths at JRC = 15.5.

No. | Group | Length of Fracture (mm) | No. | Group | Fracture Dip (°) |
---|---|---|---|---|---|

1 | l-10 | 10 | 8 | a-0 | 0 |

2 | l-20 | 20 | 9 | a-15 | 15 |

3 | l-30 | 30 | 10 | a-30 | 30 |

4 | l-40 | 40 | 11 | a-45 | 45 |

5 | l-50 | 50 | 12 | a-60 | 60 |

6 | l-60 | 60 | 13 | a-75 | 75 |

7 | l-70 | 70 | 14 | a-90 | 90 |

No. | Axial Stress σ_{1} (MPa) | Surrounding Stress σ_{2}, σ_{3} (MPa) | Seepage Pressure p_{w} (MPa) |
---|---|---|---|

1 | 6 | 4 | 3 |

2 | 6 | 5 | 3 |

3 | 6 | 6 | 3 |

4 | 8 | 4 | 3 |

5 | 8 | 5 | 3 |

6 | 8 | 6 | 3 |

7 | 10 | 4 | 3 |

8 | 10 | 5 | 3 |

9 | 10 | 6 | 3 |

Parameters | Values (in Units) | Parameters | Values (in Units) |
---|---|---|---|

Simulated object | Fracture seepage | Young’s modulus of the substrate | 4 GPa |

Size | 1 × 1~30 × 30 m | Initial porosity of the substrate | 0.25 |

Roughness factor JRC | 5.5, 10.5, 15.5 | Substrate density | 1250 kg/m^{3} |

Fracture width | 1, 1.5, 2 mm | Poisson’s ratio | 0.35 |

Normal stress | 5~20 MPa | Fluid dynamic viscosity | 0.001 Pa·s |

Category | R | R^{2} | Adjusted R^{2} | Error in Standard Estimation | Durbin–Watson |
---|---|---|---|---|---|

Experiment ^{a} | 0.809 | 0.655 | 0.628 | 1.683 × 10^{−18} | 1.931 |

Numerical simulation ^{b} | 0.932 | 0.869 | 0.868 | 5.751 × 10^{−9} | 1.198 |

Models | Unstandardised Factor | Standardisation Factor | t | Significance | Covariance Statistics | ||
---|---|---|---|---|---|---|---|

B | Standard Error | Beta | Tolerance | ||||

Experiment | (Constant) | 1.084 × 10^{−17} | 1.74 × 10^{−18} | 6.229 | 2.76 × 10^{−7} | ||

Fracture length | 7.088 × 10^{−17} | 1.726 × 10^{−17} | 0.416 | 4.108 | 2.05 × 10^{−4} | 0.884 | |

Surrounding pressure | −1.326 × 10^{−18} | 3.18 × 10^{−19} | −0.397 | −4.170 | 1.7 × 10^{−4} | 1.000 | |

Fracture dip | −3.918 × 10^{−20} | 8.932 × 10^{−21} | −0.445 | −4.386 | 8.8 × 10^{−5} | 0.884 | |

Numerical simulation | (Constant) | 4.956 × 10^{−9} | 1.726 × 10^{−9} | 2.871 | 0.004 | ||

Fracture length | −1.817 × 10^{−12} | 4.043 × 10^{−11} | −0.001 | −0.045 | 0.964 | 1.000 | |

Fracture roughness | −2.304 × 10^{−9} | 8.572 × 10^{−11} | −0.596 | −26.878 | 8.705 × 10^{−78} | 1.000 | |

Fracture width | 2.776 × 10^{−8} | 8.572 × 10^{−10} | 0.717 | 32.378 | 2.880 × 10^{−94} | 1.000 |

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## Share and Cite

**MDPI and ACS Style**

Shi, Z.; Yao, Q.; Wang, W.; Su, F.; Li, X.; Zhu, L.; Wu, C.
Size Effects of Rough Fracture Seepage in Rocks of Different Scales. *Water* **2023**, *15*, 1912.
https://doi.org/10.3390/w15101912

**AMA Style**

Shi Z, Yao Q, Wang W, Su F, Li X, Zhu L, Wu C.
Size Effects of Rough Fracture Seepage in Rocks of Different Scales. *Water*. 2023; 15(10):1912.
https://doi.org/10.3390/w15101912

**Chicago/Turabian Style**

Shi, Zhuolin, Qiangling Yao, Weinan Wang, Fengsheng Su, Xuehua Li, Liu Zhu, and Chengle Wu.
2023. "Size Effects of Rough Fracture Seepage in Rocks of Different Scales" *Water* 15, no. 10: 1912.
https://doi.org/10.3390/w15101912