Fractional Derivative Model on Physical Fractal Space: Improving Rock Permeability Analysis
Abstract
:1. Introduction
2. Permeability Measurements by Transient Pulse Testing
2.1. Traditional Perspective: Relaxation of Integer-Order Control Equations with Negative Exponentials
2.2. Fractal Perspective: Relaxation of Fractional-Order Control Equations with Mittag–Leffler Functions
2.2.1. Mechanical–Electrical Analogy of Rock Permeability Parameters
2.2.2. Abstraction of a Physical Fractal Circuit for Pore-Fracture Structures
- Branches leading to other capillaries are uniformly distributed along the current capillary’s axis.
- Only the axial resistance effect of the current capillary is considered, ignoring its water storage capacity.
- The branch capillaries of the flow path follow the same pattern.
- Parallel connection of a resistor element R and a capacitor element C to form a basic unit.
- Replication of the basic unit to create a secondary structure.
- Series connection of the secondary structure with the resistor R and capacitor C elements in the basic structure.
- Replication of this structure again to form a tertiary structure and series connection with the elements in the secondary structure.
2.2.3. Fractal Admittance Operator Based on Operator Algebra Method
2.2.4. Fractional Order Control Equations Derived from Fractal Admittance Operators
3. Experimental Validation of the Fractional Order Pulse Transient Method
3.1. Specimens and Experimental Equipment
3.2. Experimental Procedure
- (1)
- Axial and confining pressures are simultaneously increased at a rate of 10 kN/min to a hydrostatic pressure of 10 MPa.
- (2)
- The axial pressure is raised to its peak value, while the confining pressure is gradually decreased in 1 MPa steps. Permeability is measured three times before the peak stress is reached.
- (3)
- After achieving the specimen’s peak stress, a permeability measurement is conducted, and the method of axial loading is switched to displacement control at a rate of 0.06 mm/min.
- (4)
- Axial pressure is progressively decreased in 20% stress decrements, with confining pressure also being reduced in 1 MPa steps until the specimen fails completely, with permeability measured three more times after the peak stress.
3.3. Test Results and Analysis
4. Discussion
4.1. Generalized Model of Permeability Evolution
4.2. Application Scope of the Fractional Order Pulse Transient Method
4.3. Fractional Order Responses in Rock Physical Fractal Space
5. Conclusions
- (1)
- Compared to traditional integer order control equations, the fractional order control equations more accurately describe the non-Darcy flow behaviour in coals, thereby improving the precision of permeability measurements.
- (2)
- Through fitting experimental data from other rock types, such as clay and limestone, this study validates the wide applicability of the fractional order method. This suggests that fractional order derivatives hold potential advantages in describing complex seepage processes, serving as a powerful analytical tool for the fields of rock mechanics and geological engineering.
- (3)
- By investigating the relationship between the microscale fractal structure of rocks and their macroscale fractional order response, this paper lays a new theoretical foundation for the physical significance of fractional order derivatives. The extension of this theory not only deepens our understanding of the complex internal flow mechanisms in rocks, but may also facilitate the application of fractional models in rock mechanics and related domains.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Liu, Z.; Yu, X.; Xie, S.; Zhou, H.; Yin, Y. Fractional Derivative Model on Physical Fractal Space: Improving Rock Permeability Analysis. Fractal Fract. 2024, 8, 470. https://doi.org/10.3390/fractalfract8080470
Liu Z, Yu X, Xie S, Zhou H, Yin Y. Fractional Derivative Model on Physical Fractal Space: Improving Rock Permeability Analysis. Fractal and Fractional. 2024; 8(8):470. https://doi.org/10.3390/fractalfract8080470
Chicago/Turabian StyleLiu, Zelin, Xiaobin Yu, Selin Xie, Hongwei Zhou, and Yajun Yin. 2024. "Fractional Derivative Model on Physical Fractal Space: Improving Rock Permeability Analysis" Fractal and Fractional 8, no. 8: 470. https://doi.org/10.3390/fractalfract8080470