Zelin Liu

Coalbed methane is vital for the transition toward low-carbon energy systems, yet its recovery efficiency is critically limited by inaccurate classification of movable water during drainage and depressurization due to the complex pore–fracture system. To understand the influence of the pore–fracture structure on water flow law in coal reservoirs, this study constructed the relationship based on the memory effect of multiscale complex pore–fracture structures on seepage. Nuclear magnetic resonance (NMR) measurements were performed on water-saturated coal samples both before and after centrifugation, enabling the experimental identification of absolute irreducible water, partial movable water, and absolute movable water and yielding dual cutoffs. The complexity of the pore–fracture structure of the samples was quantified by multifractal analysis of the NMR test results. A fractional derivative model was developed to determine dual cutoffs, T_2c1 and T_2c2, based on the memory effect and validated against experimental data. Compared to empirical models, the proposed fractional derivative model improves R² fitting accuracy by 4.2% for T_2c1 and 9.7% for T_2c2, demonstrating its superior capability in translating structural complexity into physically meaningful cutoff determination. This work provides a mechanism-based approach for water typing, presenting a reliable foundation for drainage and depressurization in coalbed methane reservoir.

Viscoelastic materials are commonly used in civil engineering, biomedical sciences, and polymers, where understanding their creep and relaxation behaviors is essential for predicting long-term performance. This paper introduces an operator-based method for modeling viscoelastic materials, providing a unified framework to describe both creep and relaxation functions. The method utilizes stiffness and compliance operators, offering a systematic approach for analyzing viscoelastic problems. The operator-based method enhances the mathematical duality between the creep and relaxation functions, providing greater physical intuition and understanding of time-dependent material behavior. It directly reflects the intrinsic properties of materials, independent of input and output conditions. The method is extended to dynamic problems, with complex modulus and compliance derived through operator representations. The fractal tree model, with its constant loss factor across the frequency spectrum, demonstrates potential engineering applications. By incorporating a damage-based variable coefficient, the model now also accounts for the accelerated creep phase of rocks, capturing damage evolution under prolonged loading. While promising, the current method is limited to one-dimensional problems, and future research will aim to extend it to three-dimensional cases, integrate experimental validation, and explore broader applications.

Coal resource extraction and utilization are essential for sustainable development and economic growth. This study integrates a pseudo-triaxial mechanical loading system with low-field nuclear magnetic resonance (NMR) to enable the preliminary visualization of coal’s pore-fracture structure (PFS) under mechanical stress. Pseudo-triaxial and cyclic loading–unloading tests were combined with real-time NMR monitoring to model porosity recovery, pore size evolution, and energy dissipation, while also calculating the fractal dimensions of pores in relation to stress. The results show that during the compaction phase, primary pores are compressed with limited recovery after unloading. In the elastic phase, both adsorption and seepage pores transform significantly, with most recovering post-unloading. After yield stress, new fractures and pores form, and unloading enhances fracture connectivity. Seepage pore porosity shows a negative exponential relationship with axial strain before yielding, and a logarithmic relationship afterward. The fractal dimension of adsorption pores decreases during compaction and increases afterward, while the fractal dimension of seepage pores decreases before yielding and increases post-yielding. These findings provide new insights into the flow patterns of methane in coal seams.

Traditional operational calculus, while intuitive and effective in addressing problems in physical fractal spaces, often lacks the rigorous mathematical foundation needed for fractional operations, sometimes resulting in inconsistent outcomes. To address these challenges, we have developed a universal framework for defining the fractional calculus operators using the generalized fractional calculus with the Sonine kernel. In this framework, we prove that the α-th power of a differential operator corresponds precisely to the α-th fractional derivative, ensuring both accuracy and consistency. The relationship between the fractional power operators and fractional calculus is not arbitrary, it must be determined by the specific operator form and the initial conditions. Furthermore, we provide operator representations of commonly used fractional derivatives and illustrate their applications with examples of fractional power operators in physical fractal spaces. A superposition principle is also introduced to simplify fractional differential equations with non-integer exponents by transforming them into zero-initial-condition problems. This framework offers new insights into the commutative properties of fractional calculus operators and their relevance in the study of fractal structures.

As challenges in gas extraction from coal mines increase, precise measurement of permeability becomes crucial. This study proposes a novel pulse transient method based on a fractional derivative model derived on physical fractal space, incorporating operator algebra and the mechanics–electricity analogy to derive a new control equation that more accurately delineates the permeability evolution in coal. To validate the approach, permeability experiments were conducted on coal samples under mining stress conditions. The results showed that the adoption of a physically meaningful fractional-order relaxation equation provides a more accurate description of non-Darcy flow behaviour in rocks than traditional integer-order control equations. Additionally, the method proved effective across different rock types, verifying its broad applicability. By establishing a new theoretical foundation, this approach illustrates how the microscale fractal structure of rocks is fundamentally linked to their macroscale fractional responses, thereby enhancing the understanding of fractional modelling methods in rock mechanics and related domains.

An investigation into spontaneous imbibition in porous media is of paramount scientific significance in various projects. However, a precise understanding of the interaction mechanisms between media structural characteristics and imbibition remains elusive, and quantitative analysis of the interfacial interaction is lacking. Therefore, to mitigate the influence of dispersion, this study first investigates cyclic imbibition experiments of coal samples to explore the interaction mechanism between pore-fracture structure (PFS) and imbibition. Nuclear magnetic resonance is used to visualize water transport during imbibition across all scales. Subsequently, the slake durability index is suggested to clarify the coupling relationship between water–coal interactions and imbibition. Two more comprehensive and accurate imbibition models are established, based on pore size and comprehensive seepage parameters, respectively. The results demonstrate that both new models exhibit superior conformity with experimental data compared to traditional models. The memory factor quantifies interface interaction within these models. Sensitivity analysis reveals that strong interface interaction diminishes the effective imbibition ratio, while the structural characteristics of porous media significantly influence the interaction. Furthermore, the fractal dimension quantitatively characterizes the PFS features of coal samples. An exploration of the relationship between fractal dimension and memory factor indicates the influence of porous media heterogeneity on imbibition.