Fractal and Fractional

This paper studies mixed impulsive boundary value problems involving tempered $\mathsf{\Psi }$-fractional derivatives of Caputo type. By introducing exponential tempering into the fractional framework, the proposed model effectively captures systems with fading memory—an improvement over conventional power-law kernels that assume long-range dependence. The generalized tempered $\mathsf{\Psi }$-operator unifies several existing fractional derivatives, offering enhanced flexibility for modeling complex dynamical phenomena. Impulsive effects and integral boundary conditions are incorporated to describe processes subject to sudden changes and historical dependence. The problem is reformulated as a Volterra integral equation, and fixed-point theory is employed to establish analytical results. Existence and uniqueness of solutions are proven using the Banach Contraction Mapping Principle, while the Leray–Schauder nonlinear alternative ensures existence in non-contractive cases. The proposed framework provides a rigorous analytical basis for modeling phenomena characterized by both fading memory and sudden perturbations, with potential applications in physics, control theory, population dynamics, and epidemiology. A numerical example is presented to illustrate the validity and applicability of the main theoretical results.

The Jurassic lacustrine oil shale in southwest China has become a primary production layer due to its high yield and substantial reserves. However, influenced by the lacustrine environment, the vertical profile of the lacustrine shale reservoir shows alternating deposits of shale and carbonate rock. This complex lithological combination results in significant heterogeneity in reservoir types, reservoir distribution, and internal structure. Currently, research on micro-pore structure and hydrocarbon storage mechanisms in lacustrine shales is insufficient, necessitating the elucidation of their micro-characteristics to support future exploration and development. This research focuses on the Da’anzhai Member of Jurassic Ziliujing Formation. Various techniques—including organic geochemical analysis, X-ray diffraction, physical property testing, gradient centrifugation, and gradient drying NMR monitoring—were employed to investigate the micro-pore structure and fluid storage mechanisms of the lacustrine shale reservoir. The following insights were gained from this research. The organic matter pores (OMP) and inorganic pores (IP) developed within the Da’anzhai lacustrine shale reservoir together create the storage space for shale oil, while micro-fractures further enhance the reservoir’s storage capacity and flow performance. Lacustrine shale oil exists in three storage states: mobile oil, bound oil, and adsorbed oil. Mobile oil is primarily located within the micro-fractures and large pores (greater than 350 nm) of the shale reservoir and is the main target for industrial extraction. Bound oil is mainly found in the meso-pores, micropores, and narrow pore structures between rock grains (30 nm to 350 nm), and, theoretically, could potentially be developed through engineering methods such as hydraulic fracturing. Adsorbed oil, due to its close binding with organic matter and clay mineral surfaces, is difficult to release effectively using conventional techniques. The OM abundance, the mineral composition of lacustrine shale, and the pore structure all influence the storage states of shale oil. While a high TOC value increases the amount of mobile oil, the strong adsorption properties of kerogen and organic matter lead to the accumulation of adsorbed oil, which inhibits oil flow. Clay minerals further restrict oil flow by enhancing adsorption, while brittle minerals facilitate the movement of mobile oil by expanding pore space. Based on fractal geometry theory and multi-scale testing results, the large pores in the Da’anzhai lacustrine shale have a high fractal dimension and exhibit complex shapes. However, as pore complexity increases, the amount of adsorbed oil rises significantly, which in turn reduces the proportion of movable oil.

During coal mining, the development of joint fractures in overlying rock strata is one of the key factors that degrade the mechanical properties of rock masses, form water-conducting fracture zones, and induce safety hazards. To investigate the fracture evolution characteristics of overlying strata during coal extraction under thick and hard roof conditions, this study established a mining physical model based on similarity simulation technology, tracked the fracture evolution process, and performed quantitative analysis using fractal theory. The results show that fracture development is significantly correlated with the mining advance distance: the fractal dimension of fractures is small in the initial mining stage and gradually increases as the working face advances. When the mining width exceeds the ultimate span of the roof, local fractures expand rapidly with a sharp rise in the fractal dimension to 1.436; further increasing the mining width triggers large-scale sudden fracture expansion, resulting in severe degradation of rock mass integrity, with the maximum fractal dimension reaching 1.445. The research findings provide theoretical references for safety management and disaster prevention in coal mining under thick and hard roof conditions.