Investigation of Power-Law Fluid Infiltration Grout Characteristics on the Basis of Fractal Theory
Abstract
:1. Introduction
2. Modeling of Power-Law Fluid Diffusion Fractal Infiltration Grouting
2.1. Fractal Modeling of Porous Media
2.2. Fractal Dimension Power-Law Fluid Percolation Equation
2.3. Power-Law Fluid Spherical Diffusion Fractal Infiltration Grouting Model
2.4. Power-Law Fluid Column Diffusion Fractal Infiltration Grouting Model
2.5. Scope of Application of the Formula and Discussion
2.5.1. Scope of Use of the Formula
- (1)
- The fluid movement within the formation is characterized by laminar flow, with a Reynolds number ;
- (2)
- The impact of the slurry’s self-weight is disregarded during the grouting process;
- (3)
- The injected porous medium consists of isotropic material, and the grouting process employs constant-pressure grouting;
- (4)
- In the context of a porous medium, the fractal dimension must adhere to specific constraints: and .
2.5.2. Solving for the Fractal Dimension
2.5.3. Discussion of the Diffusion Equation
3. Study of the Laws of Diffusion
4. Indoor Grouting Experiment
4.1. Experimental Setup and Grouting Materials
4.2. Determination of Fractal Parameters and Design of Grouting
- A high-resolution image of the injected medium is imported into the software and subsequently converted to a black-and-white format.
- The software’s color range feature is utilized to select the sample pores (Figure 9).
- The analysis tool is utilized to record measurements and identify the pixel area with the largest area in all the black ranges. The conversion results in . The minimum pore diameter differs from the maximum pore diameter by three orders of magnitude [48]. can be calculated using .
4.3. Experiments and Analysis of Results
- Neglect of the effect of the pressure-loading dynamic process
- 2.
- Geometric constraint and gravity coupling effect
- 3.
- Multi-scale uncertainty sources
5. Numerical Modeling Study
5.1. Creation of Models
5.2. Simulation Results
5.3. Numerical Simulation Results
6. Conclusions
- The fractal permeability constant exhibits an exponential relationship with the rheological index, demonstrating a threshold effect on its sensitivity. When the rheological index, n, is below 0.4, the influence coefficient of the power index on the fractal permeability is minimal and the fractal permeability constant approaches 0. However, when n exceeds 0.4, the sensitivity increases significantly, with the fractal permeability constant reaching 0.48 at n = 0.9. In the diffusion model study, the pulp diffusion radius shows a non-linear proportional relationship to the slurry pressure and the water–cement ratio. At pressures below 0.1 MPa, the diffusion distance ratio between the spherical and columnar geometries is 1:0.96, indicating limited geometric constraint effects in low-energy grouting. At P = 0.5 MPa, the isotropic advantage of spherical diffusion becomes apparent, with the diffusion distance ratio expanding to 1:0.82. As the water–cement ratio increases from 0.5 to 0.7, the sensitivity of spherical grouting under high water–cement ratios significantly increases, enhancing the penetration efficiency advantage of spherical grouting. Mechanistically, the increase in the water–cement ratio leads to a synergistic decrease in slurry viscosity and yield stress, weakening the shear resistance at the slurry–soil interface and promoting seepage diffusion. This finding aligns closely with the use of high-water–cement ratio slurries for long-distance fissure sealing in engineering practice.
- In experimental and theoretical studies, the theoretical–experimental error of spherical grouting converged from the initial 21.54% (t = 5 s) to 8.43% (t = 15 s) by exponential decay, while the error of column grouting decreased non-linearly from 25.45% to 10.17%. Despite the above deviations, the errors were located within the engineering tolerance error band (−50%~100%), which proves the engineering applicability of the model. The errors were analyzed mainly from three aspects, neglecting the effect of the pressure-loading dynamic process, geometric constraint and gravity coupling effects, and multi-scale uncertainty sources.
- Numerical simulation studies reveal that as grouting time increases, the pressure and slurry flow rate of power-law fluid columnar and spherical grouting continuously spread outward. The maximum grouting pressure reaches 0.4 MPa, while the maximum flow rate for both types is 2 m/s. Comparing the meander fractal dimension with the volume fractal dimension, it was observed that when the meander fractal dimension was 1.0, the columnar diffusion distance was 168 mm and the spherical diffusion distance was 216 mm. When the volume fractal dimension was 2.8, the columnar diffusion distance extended to 192 mm and the spherical diffusion distance reached 229 mm. The difference between these two scenarios (48 mm for the meander fractal dimension and 37 mm for the volume fractal dimension) indicates that the meander fractal dimension parameters are more sensitive. Furthermore, comparing spherical and columnar diffusion under the same fractal dimension demonstrates that spherical diffusion consistently exhibits a larger range than columnar diffusion.
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Glossary
Symbols | Definitions |
Equivalent diameter | |
Total cumulative number of pore areas | |
Maximum pore diameter | |
Volume fractal dimension | |
Degree of tortuosity | |
Effective length of pores | |
Tortuous fractal dimension | |
Pore length | |
τ | Shear stress |
Coefficient of consistency | |
Rheological index | |
Apparent viscosity | |
Fluid flow | |
Effective pore length | |
Reference area | |
Fractal permeability constant | |
Effective volume of grouting | |
Total volume of the sphere model | |
Porosity | |
Spherical modeling fluid flow | |
Column modeled fluid flow | |
Average flow rate of fluid in a pore | |
Average fluid flow rate | |
Ball model grouting area | |
Area of column model grouting | |
Grouting pipe radius | |
Slurry diffusion radius | |
Grouting pressure | |
Water head pressure at grouting point | |
d | Spatial geometric dimension |
τav | Average tortuosity |
λav | Average capillary diameter |
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Rheological Characteristics | |||
---|---|---|---|
Coefficient of consistency | 6.475 | 3.246 | 1.569 |
Rheological index | 0.324 | 0.330 | 0.308 |
Apparent viscosity | 0.314 | 0.072 | 0.028 |
Rheological equation |
Parameter Category | Coarse Sand | |
---|---|---|
Experimental measurement index | Porosity, ϕ/ % | 41.80 |
Maximum pore diameter, | 0.43 | |
Derived calculated indicators | Average tortuosity, τav | 2.07 |
Average capillary diameter, | 0.0072 | |
Volume dimension, Df | 2.4596 |
Serial Number | Experimental Values | Theoretical Values | Difference Analysis % | |||
---|---|---|---|---|---|---|
Measured Grouting Pressure Value | Grouting Time | Experimental Diffusion Radius | Tortuous Fractal Dimension (Math.) | Calculated Diffusion Grouting | ||
K1 | 0.418 | 5 | 65 | 1.1025 | 48 | 26.15 |
K2 | 0.389 | 10 | 144 | 1.0995 | 128 | 11.11 |
K3 | 0.394 | 15 | 182 | 1.0977 | 167 | 8.43 |
S1 | 0.392 | 5 | 55 | 1.0700 | 41 | 25.45 |
S2 | 0.408 | 10 | 84 | 1.0682 | 71 | 15.48 |
S3 | 0.407 | 15 | 118 | 1.0670 | 106 | 10.17 |
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Wei, F.; Lai, J.; Su, X. Investigation of Power-Law Fluid Infiltration Grout Characteristics on the Basis of Fractal Theory. Buildings 2025, 15, 987. https://doi.org/10.3390/buildings15060987
Wei F, Lai J, Su X. Investigation of Power-Law Fluid Infiltration Grout Characteristics on the Basis of Fractal Theory. Buildings. 2025; 15(6):987. https://doi.org/10.3390/buildings15060987
Chicago/Turabian StyleWei, Fucheng, Jinxing Lai, and Xulin Su. 2025. "Investigation of Power-Law Fluid Infiltration Grout Characteristics on the Basis of Fractal Theory" Buildings 15, no. 6: 987. https://doi.org/10.3390/buildings15060987
APA StyleWei, F., Lai, J., & Su, X. (2025). Investigation of Power-Law Fluid Infiltration Grout Characteristics on the Basis of Fractal Theory. Buildings, 15(6), 987. https://doi.org/10.3390/buildings15060987