Fractal Algorithm and Its Application in Rock Mechanics

Contents
1 Fractal Algorithm of an Area Covering Method for Describing the Self-Similarity of Spatial Curve
	1.1 Introduction
	1.2 Structure of 3D Fractal Curve and Its Fractal Dimension
	1.3 Area Covering Method
	1.4 Calculation of Fractal Dimension of Spatial Fractal Curve
	1.5 Conclusion
2 Fractal Algorithm of a Volume Covering Method for Describing the Self-Similarity of Curved and Rough Surface
	2.1 Introduction
	2.2 Preparation of Rock Samples
	2.3 Prism-Like Volume Covering Method
	2.4 3-Sided Parallel Body Volume Covering Method
	2.5 Comparative Analysis of Different Methods
	2.6 Conclusion
3 Fractal Algorithm of Recognization and Reconstruction of Pore Structure Using AI Technology
	3.1 Introduction
	3.2 Basic Principles of Fully Convolutional Neural Networks
	3.3 Micro CT-Based Fracture Network Reconstruction
		3.3.1 Micro CT Scanning Test
		3.3.2 Reconstruction of Pore Structure in Coal
		3.3.3 Quantitative Statistics of Pores in Coal
	3.4 FCN Algorithm and Its Structure
		3.4.1 Establishment of Data Sets for Recognization
		3.4.2 Architecture for FCN
		3.4.3 Data Extraction of Fracture Geometry and Network Topology
		3.4.4 Quantitative Extraction of Topological Geometry of Fracture Network
		3.4.5 Modification of Topological Geometry of Fracture Network
	3.5 CT Slices and 3D Reconstruction
		3.5.1 Material Phase Definition and Reconstruction
		3.5.2 FCN-Based 3D Reconstruction
	3.6 Three-Dimensional Fractal Reconstruction
	3.7 Conclusion
4 First Application in Granite Drilling to Evaluate the Statistics of Joint Space Using Multifractal Dimension
	4.1 Introduction
	4.2 Statistics and Revision of the Number of Joints from Core Drilling
		4.2.1 Statistical Analysis of Joint Occurrence in a Drilling Core
		4.2.2 Statistical Analysis of Joint Spacing in Drilling Hole
		4.2.3 Evaluation of Quality of Deep Rock
	4.3 Multifractal Characteristics of Spatial Distribution for Joints
		4.3.1 Multifractal Spectrum of Joint Number and Spacing
		4.3.2 Multifractal Spectrum of Joint Number and Depth
	4.4 Conclusion
5 Second Application in Low-Permeability Coal to Describe the Motion Equation of Interface in Gas–Water Flow Using a Fractal Model
	5.1 Introduction
	5.2 Derivation of Tortuosity from Hagen–Poiseuille Equation
	5.3 Calculation of Tortuosity of Fractal Pores in Low Permeability Coal
	5.4 Derivation of Fractal Dimension of the Sectional Profile of Capillary
	5.5 Fractal Motion Equation of Gas–Liquid Flow
	5.6 Analysis of Fractal Motion Equation in the Experiment
	5.7 Conclusion and Discussion
6 Third Application in Coal to Describe Experimental Tortuosity Using a Fractal Model
	6.1 Coal Sample Preparation and CT Scanning Device
		6.1.1 Coal Samples
		6.1.2 Experimental Device
	6.2 Determination of Threshold of CT Slice for Binary Segmentation
	6.3 3D Reconstruction of Micro- and Nano-Pore Network
		6.3.1 Analysis of Nano-CT-Based Test Results
		6.3.2 Analysis of Micron CT Test Results
	6.4 Research on Pore Distribution Based on Mercury Injection Method
	6.5 Comparison of Pore Distribution by Three Methods
	6.6 Conclusion
7 Fourth Application in Coal to Describe Theoretical Tortuosity Using a Fractal Model
	7.1 Definition of Tortuosity in Medium
	7.2 Estimation of Capillary Tortuosity and Fractal Dimension
	7.3 Dependent Analysis of Permeability on Tortuosity
	7.4 Calculation of Fractal Dimension Based on Nano CT Slices
	7.5 Conclusion
8 Fifth Application in Low-Permeability Salt-Rock Under Thermal–mechanical Coupling to Describe the Long-Term Creep Behavior Using Fractional Model
	8.1 Introduction
	8.2 Damage Definition Considering Temperature and Volume Stress
	8.3 Establishment of the Fractional Creep Damage Burgers Model
		8.3.1 Definition of Fractional Calculus
		8.3.2 Establishment of Fractional Abel Dashpot
		8.3.3 Establishment of Fractional Nonlinear Dashpot
		8.3.4 Theoretical Solution of Fractional Creep Damage Burgers Model
	8.4 Verification of Fractional Creep Damage Burgers Model
		8.4.1 Model Verification Under Different Temperatures
		8.4.2 Model Validation Under Different Volume Stresses
		8.4.3 Sensitivity of Temperature on Modeling Parameters
	8.5 Prediction of Accelerated Creep Deformation by Fractional Burgers Model
		8.5.1 Prediction of Accelerated Creep at Different Temperatures
		8.5.2 Prediction of Accelerated Creep Under Different Volume Stresses
	8.6 Conclusion
References